Numeric.SpecFunctions:logGamma from math-functions-0.1.5.2, A

Percentage Accurate: 100.0% → 100.0%

Time: 7.0s

Alternatives: 12

Speedup: 1.0×

Specification

Math

\[\begin{array}{l} \\ \left(x \cdot \left(y - 1\right) - y \cdot 0.5\right) + 0.918938533204673 \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (+ (- (* x (- y 1.0)) (* y 0.5)) 0.918938533204673))

C

double code(double x, double y) {
	return ((x * (y - 1.0)) - (y * 0.5)) + 0.918938533204673;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    code = ((x * (y - 1.0d0)) - (y * 0.5d0)) + 0.918938533204673d0
end function

Java

public static double code(double x, double y) {
	return ((x * (y - 1.0)) - (y * 0.5)) + 0.918938533204673;
}

Python

def code(x, y):
	return ((x * (y - 1.0)) - (y * 0.5)) + 0.918938533204673

Julia

function code(x, y)
	return Float64(Float64(Float64(x * Float64(y - 1.0)) - Float64(y * 0.5)) + 0.918938533204673)
end

MATLAB

function tmp = code(x, y)
	tmp = ((x * (y - 1.0)) - (y * 0.5)) + 0.918938533204673;
end

Wolfram

code[x_, y_] := N[(N[(N[(x * N[(y - 1.0), $MachinePrecision]), $MachinePrecision] - N[(y * 0.5), $MachinePrecision]), $MachinePrecision] + 0.918938533204673), $MachinePrecision]

Initial Program: 100.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \left(x \cdot \left(y - 1\right) - y \cdot 0.5\right) + 0.918938533204673 \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (+ (- (* x (- y 1.0)) (* y 0.5)) 0.918938533204673))

C

double code(double x, double y) {
	return ((x * (y - 1.0)) - (y * 0.5)) + 0.918938533204673;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    code = ((x * (y - 1.0d0)) - (y * 0.5d0)) + 0.918938533204673d0
end function

Java

public static double code(double x, double y) {
	return ((x * (y - 1.0)) - (y * 0.5)) + 0.918938533204673;
}

Python

def code(x, y):
	return ((x * (y - 1.0)) - (y * 0.5)) + 0.918938533204673

Julia

function code(x, y)
	return Float64(Float64(Float64(x * Float64(y - 1.0)) - Float64(y * 0.5)) + 0.918938533204673)
end

MATLAB

function tmp = code(x, y)
	tmp = ((x * (y - 1.0)) - (y * 0.5)) + 0.918938533204673;
end

Wolfram

code[x_, y_] := N[(N[(N[(x * N[(y - 1.0), $MachinePrecision]), $MachinePrecision] - N[(y * 0.5), $MachinePrecision]), $MachinePrecision] + 0.918938533204673), $MachinePrecision]

Alternative 1: 100.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \left(x \cdot \left(y + -1\right) - y \cdot 0.5\right) + 0.918938533204673 \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (+ (- (* x (+ y -1.0)) (* y 0.5)) 0.918938533204673))

C

double code(double x, double y) {
	return ((x * (y + -1.0)) - (y * 0.5)) + 0.918938533204673;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    code = ((x * (y + (-1.0d0))) - (y * 0.5d0)) + 0.918938533204673d0
end function

Java

public static double code(double x, double y) {
	return ((x * (y + -1.0)) - (y * 0.5)) + 0.918938533204673;
}

Python

def code(x, y):
	return ((x * (y + -1.0)) - (y * 0.5)) + 0.918938533204673

Julia

function code(x, y)
	return Float64(Float64(Float64(x * Float64(y + -1.0)) - Float64(y * 0.5)) + 0.918938533204673)
end

MATLAB

function tmp = code(x, y)
	tmp = ((x * (y + -1.0)) - (y * 0.5)) + 0.918938533204673;
end

Wolfram

code[x_, y_] := N[(N[(N[(x * N[(y + -1.0), $MachinePrecision]), $MachinePrecision] - N[(y * 0.5), $MachinePrecision]), $MachinePrecision] + 0.918938533204673), $MachinePrecision]

Derivation

1. Initial program 100.0%

   \[\left(x \cdot \left(y - 1\right) - y \cdot 0.5\right) + 0.918938533204673 \]

2. Final simplification 100.0%

   \[\leadsto \left(x \cdot \left(y + -1\right) - y \cdot 0.5\right) + 0.918938533204673 \]

Alternative 2: 49.9% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -3.6 \cdot 10^{+153}:\\ \;\;\;\;-x\\ \mathbf{elif}\;x \leq -39000:\\ \;\;\;\;x \cdot y\\ \mathbf{elif}\;x \leq 0.92:\\ \;\;\;\;0.918938533204673\\ \mathbf{elif}\;x \leq 3 \cdot 10^{+130} \lor \neg \left(x \leq 5.2 \cdot 10^{+222}\right) \land x \leq 7.2 \cdot 10^{+279}:\\ \;\;\;\;-x\\ \mathbf{else}:\\ \;\;\;\;x \cdot y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x -3.6e+153)
   (- x)
   (if (<= x -39000.0)
     (* x y)
     (if (<= x 0.92)
       0.918938533204673
       (if (or (<= x 3e+130) (and (not (<= x 5.2e+222)) (<= x 7.2e+279)))
         (- x)
         (* x y))))))

C

double code(double x, double y) {
	double tmp;
	if (x <= -3.6e+153) {
		tmp = -x;
	} else if (x <= -39000.0) {
		tmp = x * y;
	} else if (x <= 0.92) {
		tmp = 0.918938533204673;
	} else if ((x <= 3e+130) || (!(x <= 5.2e+222) && (x <= 7.2e+279))) {
		tmp = -x;
	} else {
		tmp = x * y;
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (x <= (-3.6d+153)) then
        tmp = -x
    else if (x <= (-39000.0d0)) then
        tmp = x * y
    else if (x <= 0.92d0) then
        tmp = 0.918938533204673d0
    else if ((x <= 3d+130) .or. (.not. (x <= 5.2d+222)) .and. (x <= 7.2d+279)) then
        tmp = -x
    else
        tmp = x * y
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (x <= -3.6e+153) {
		tmp = -x;
	} else if (x <= -39000.0) {
		tmp = x * y;
	} else if (x <= 0.92) {
		tmp = 0.918938533204673;
	} else if ((x <= 3e+130) || (!(x <= 5.2e+222) && (x <= 7.2e+279))) {
		tmp = -x;
	} else {
		tmp = x * y;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if x <= -3.6e+153:
		tmp = -x
	elif x <= -39000.0:
		tmp = x * y
	elif x <= 0.92:
		tmp = 0.918938533204673
	elif (x <= 3e+130) or (not (x <= 5.2e+222) and (x <= 7.2e+279)):
		tmp = -x
	else:
		tmp = x * y
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (x <= -3.6e+153)
		tmp = Float64(-x);
	elseif (x <= -39000.0)
		tmp = Float64(x * y);
	elseif (x <= 0.92)
		tmp = 0.918938533204673;
	elseif ((x <= 3e+130) || (!(x <= 5.2e+222) && (x <= 7.2e+279)))
		tmp = Float64(-x);
	else
		tmp = Float64(x * y);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= -3.6e+153)
		tmp = -x;
	elseif (x <= -39000.0)
		tmp = x * y;
	elseif (x <= 0.92)
		tmp = 0.918938533204673;
	elseif ((x <= 3e+130) || (~((x <= 5.2e+222)) && (x <= 7.2e+279)))
		tmp = -x;
	else
		tmp = x * y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[x, -3.6e+153], (-x), If[LessEqual[x, -39000.0], N[(x * y), $MachinePrecision], If[LessEqual[x, 0.92], 0.918938533204673, If[Or[LessEqual[x, 3e+130], And[N[Not[LessEqual[x, 5.2e+222]], $MachinePrecision], LessEqual[x, 7.2e+279]]], (-x), N[(x * y), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if x < -3.6000000000000001e153 or 0.92000000000000004 < x < 2.9999999999999999e130 or 5.2000000000000002e222 < x < 7.20000000000000012e279

   1. Initial program 99.9%

      \[\left(x \cdot \left(y - 1\right) - y \cdot 0.5\right) + 0.918938533204673 \]
   2. Step-by-step derivation

      1. cancel-sign-sub-inv 99.9%

         \[\leadsto \color{blue}{\left(x \cdot \left(y - 1\right) + \left(-y\right) \cdot 0.5\right)} + 0.918938533204673 \]

      2. +-commutative 99.9%

         \[\leadsto \color{blue}{\left(\left(-y\right) \cdot 0.5 + x \cdot \left(y - 1\right)\right)} + 0.918938533204673 \]

      3. distribute-lft-out-- 100.0%

         \[\leadsto \left(\left(-y\right) \cdot 0.5 + \color{blue}{\left(x \cdot y - x \cdot 1\right)}\right) + 0.918938533204673 \]

      4. cancel-sign-sub-inv 100.0%

         \[\leadsto \left(\left(-y\right) \cdot 0.5 + \color{blue}{\left(x \cdot y + \left(-x\right) \cdot 1\right)}\right) + 0.918938533204673 \]

      5. *-rgt-identity 100.0%

         \[\leadsto \left(\left(-y\right) \cdot 0.5 + \left(x \cdot y + \color{blue}{\left(-x\right)}\right)\right) + 0.918938533204673 \]

      6. associate-+r+ 100.0%

         \[\leadsto \color{blue}{\left(\left(\left(-y\right) \cdot 0.5 + x \cdot y\right) + \left(-x\right)\right)} + 0.918938533204673 \]

      7. associate-+l+ 100.0%

         \[\leadsto \color{blue}{\left(\left(-y\right) \cdot 0.5 + x \cdot y\right) + \left(\left(-x\right) + 0.918938533204673\right)} \]

      8. +-commutative 100.0%

         \[\leadsto \left(\left(-y\right) \cdot 0.5 + x \cdot y\right) + \color{blue}{\left(0.918938533204673 + \left(-x\right)\right)} \]

      9. unsub-neg 100.0%

         \[\leadsto \left(\left(-y\right) \cdot 0.5 + x \cdot y\right) + \color{blue}{\left(0.918938533204673 - x\right)} \]

      10. associate-+r- 100.0%

          \[\leadsto \color{blue}{\left(\left(\left(-y\right) \cdot 0.5 + x \cdot y\right) + 0.918938533204673\right) - x} \]

      11. *-commutative 100.0%

          \[\leadsto \left(\left(\left(-y\right) \cdot 0.5 + \color{blue}{y \cdot x}\right) + 0.918938533204673\right) - x \]

      12. distribute-lft-neg-out 100.0%

          \[\leadsto \left(\left(\color{blue}{\left(-y \cdot 0.5\right)} + y \cdot x\right) + 0.918938533204673\right) - x \]

      13. distribute-rgt-neg-in 100.0%

          \[\leadsto \left(\left(\color{blue}{y \cdot \left(-0.5\right)} + y \cdot x\right) + 0.918938533204673\right) - x \]

      14. distribute-lft-out 100.0%

          \[\leadsto \left(\color{blue}{y \cdot \left(\left(-0.5\right) + x\right)} + 0.918938533204673\right) - x \]

      15. fma-def 100.0%

          \[\leadsto \color{blue}{\mathsf{fma}\left(y, \left(-0.5\right) + x, 0.918938533204673\right)} - x \]

      16. +-commutative 100.0%

          \[\leadsto \mathsf{fma}\left(y, \color{blue}{x + \left(-0.5\right)}, 0.918938533204673\right) - x \]

      17. sub-neg 100.0%

          \[\leadsto \mathsf{fma}\left(y, \color{blue}{x - 0.5}, 0.918938533204673\right) - x \]

   3. Simplified 100.0%

      \[\leadsto \color{blue}{\mathsf{fma}\left(y, x - 0.5, 0.918938533204673\right) - x} \]
   4. Step-by-step derivation

      1. fma-udef 100.0%

         \[\leadsto \color{blue}{\left(y \cdot \left(x - 0.5\right) + 0.918938533204673\right)} - x \]

      2. *-un-lft-identity 100.0%

         \[\leadsto \left(y \cdot \color{blue}{\left(1 \cdot \left(x - 0.5\right)\right)} + 0.918938533204673\right) - x \]

      3. *-un-lft-identity 100.0%

         \[\leadsto \left(y \cdot \color{blue}{\left(x - 0.5\right)} + 0.918938533204673\right) - x \]

      4. sub-neg 100.0%

         \[\leadsto \left(y \cdot \color{blue}{\left(x + \left(-0.5\right)\right)} + 0.918938533204673\right) - x \]

      5. metadata-eval 100.0%

         \[\leadsto \left(y \cdot \left(x + \color{blue}{-0.5}\right) + 0.918938533204673\right) - x \]

   5. Applied egg-rr 100.0%

      \[\leadsto \color{blue}{\left(y \cdot \left(x + -0.5\right) + 0.918938533204673\right)} - x \]

   6. Taylor expanded in x around inf 99.0%

      \[\leadsto \color{blue}{\left(y - 1\right) \cdot x} \]

   7. Taylor expanded in y around 0 63.6%

      \[\leadsto \color{blue}{-1 \cdot x} \]
   8. Step-by-step derivation

      1. neg-mul-1 63.6%

         \[\leadsto \color{blue}{-x} \]

   9. Simplified 63.6%

      \[\leadsto \color{blue}{-x} \]

   if -3.6000000000000001e153 < x < -39000 or 2.9999999999999999e130 < x < 5.2000000000000002e222 or 7.20000000000000012e279 < x

   1. Initial program 100.0%

      \[\left(x \cdot \left(y - 1\right) - y \cdot 0.5\right) + 0.918938533204673 \]
   2. Step-by-step derivation

      1. cancel-sign-sub-inv 100.0%

         \[\leadsto \color{blue}{\left(x \cdot \left(y - 1\right) + \left(-y\right) \cdot 0.5\right)} + 0.918938533204673 \]

      2. +-commutative 100.0%

         \[\leadsto \color{blue}{\left(\left(-y\right) \cdot 0.5 + x \cdot \left(y - 1\right)\right)} + 0.918938533204673 \]

      3. distribute-lft-out-- 100.0%

         \[\leadsto \left(\left(-y\right) \cdot 0.5 + \color{blue}{\left(x \cdot y - x \cdot 1\right)}\right) + 0.918938533204673 \]

      4. cancel-sign-sub-inv 100.0%

         \[\leadsto \left(\left(-y\right) \cdot 0.5 + \color{blue}{\left(x \cdot y + \left(-x\right) \cdot 1\right)}\right) + 0.918938533204673 \]

      5. *-rgt-identity 100.0%

         \[\leadsto \left(\left(-y\right) \cdot 0.5 + \left(x \cdot y + \color{blue}{\left(-x\right)}\right)\right) + 0.918938533204673 \]

      6. associate-+r+ 100.0%

         \[\leadsto \color{blue}{\left(\left(\left(-y\right) \cdot 0.5 + x \cdot y\right) + \left(-x\right)\right)} + 0.918938533204673 \]

      7. associate-+l+ 100.0%

         \[\leadsto \color{blue}{\left(\left(-y\right) \cdot 0.5 + x \cdot y\right) + \left(\left(-x\right) + 0.918938533204673\right)} \]

      8. +-commutative 100.0%

         \[\leadsto \left(\left(-y\right) \cdot 0.5 + x \cdot y\right) + \color{blue}{\left(0.918938533204673 + \left(-x\right)\right)} \]

      9. unsub-neg 100.0%

         \[\leadsto \left(\left(-y\right) \cdot 0.5 + x \cdot y\right) + \color{blue}{\left(0.918938533204673 - x\right)} \]

      10. associate-+r- 100.0%

          \[\leadsto \color{blue}{\left(\left(\left(-y\right) \cdot 0.5 + x \cdot y\right) + 0.918938533204673\right) - x} \]

      11. *-commutative 100.0%

          \[\leadsto \left(\left(\left(-y\right) \cdot 0.5 + \color{blue}{y \cdot x}\right) + 0.918938533204673\right) - x \]

      12. distribute-lft-neg-out 100.0%

          \[\leadsto \left(\left(\color{blue}{\left(-y \cdot 0.5\right)} + y \cdot x\right) + 0.918938533204673\right) - x \]

      13. distribute-rgt-neg-in 100.0%

          \[\leadsto \left(\left(\color{blue}{y \cdot \left(-0.5\right)} + y \cdot x\right) + 0.918938533204673\right) - x \]

      14. distribute-lft-out 100.0%

          \[\leadsto \left(\color{blue}{y \cdot \left(\left(-0.5\right) + x\right)} + 0.918938533204673\right) - x \]

      15. fma-def 100.0%

          \[\leadsto \color{blue}{\mathsf{fma}\left(y, \left(-0.5\right) + x, 0.918938533204673\right)} - x \]

      16. +-commutative 100.0%

          \[\leadsto \mathsf{fma}\left(y, \color{blue}{x + \left(-0.5\right)}, 0.918938533204673\right) - x \]

      17. sub-neg 100.0%

          \[\leadsto \mathsf{fma}\left(y, \color{blue}{x - 0.5}, 0.918938533204673\right) - x \]

   3. Simplified 100.0%

      \[\leadsto \color{blue}{\mathsf{fma}\left(y, x - 0.5, 0.918938533204673\right) - x} \]
   4. Step-by-step derivation

      1. fma-udef 100.0%

         \[\leadsto \color{blue}{\left(y \cdot \left(x - 0.5\right) + 0.918938533204673\right)} - x \]

      2. *-un-lft-identity 100.0%

         \[\leadsto \left(y \cdot \color{blue}{\left(1 \cdot \left(x - 0.5\right)\right)} + 0.918938533204673\right) - x \]

      3. *-un-lft-identity 100.0%

         \[\leadsto \left(y \cdot \color{blue}{\left(x - 0.5\right)} + 0.918938533204673\right) - x \]

      4. sub-neg 100.0%

         \[\leadsto \left(y \cdot \color{blue}{\left(x + \left(-0.5\right)\right)} + 0.918938533204673\right) - x \]

      5. metadata-eval 100.0%

         \[\leadsto \left(y \cdot \left(x + \color{blue}{-0.5}\right) + 0.918938533204673\right) - x \]

   5. Applied egg-rr 100.0%

      \[\leadsto \color{blue}{\left(y \cdot \left(x + -0.5\right) + 0.918938533204673\right)} - x \]

   6. Taylor expanded in x around inf 100.0%

      \[\leadsto \color{blue}{\left(y - 1\right) \cdot x} \]

   7. Taylor expanded in y around inf 68.0%

      \[\leadsto \color{blue}{y \cdot x} \]

   if -39000 < x < 0.92000000000000004

   1. Initial program 100.0%

      \[\left(x \cdot \left(y - 1\right) - y \cdot 0.5\right) + 0.918938533204673 \]
   2. Step-by-step derivation

      1. associate-+l- 100.0%

         \[\leadsto \color{blue}{x \cdot \left(y - 1\right) - \left(y \cdot 0.5 - 0.918938533204673\right)} \]

      2. sub-neg 100.0%

         \[\leadsto x \cdot \color{blue}{\left(y + \left(-1\right)\right)} - \left(y \cdot 0.5 - 0.918938533204673\right) \]

      3. metadata-eval 100.0%

         \[\leadsto x \cdot \left(y + \color{blue}{-1}\right) - \left(y \cdot 0.5 - 0.918938533204673\right) \]

   3. Simplified 100.0%

      \[\leadsto \color{blue}{x \cdot \left(y + -1\right) - \left(y \cdot 0.5 - 0.918938533204673\right)} \]

   4. Taylor expanded in y around inf 97.1%

      \[\leadsto \color{blue}{y \cdot x} - \left(y \cdot 0.5 - 0.918938533204673\right) \]

   5. Taylor expanded in y around 0 47.6%

      \[\leadsto \color{blue}{0.918938533204673} \]
3. Recombined 3 regimes into one program.

4. Final simplification 56.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -3.6 \cdot 10^{+153}:\\ \;\;\;\;-x\\ \mathbf{elif}\;x \leq -39000:\\ \;\;\;\;x \cdot y\\ \mathbf{elif}\;x \leq 0.92:\\ \;\;\;\;0.918938533204673\\ \mathbf{elif}\;x \leq 3 \cdot 10^{+130} \lor \neg \left(x \leq 5.2 \cdot 10^{+222}\right) \land x \leq 7.2 \cdot 10^{+279}:\\ \;\;\;\;-x\\ \mathbf{else}:\\ \;\;\;\;x \cdot y\\ \end{array} \]

Alternative 3: 98.8% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -4.5 \lor \neg \left(x \leq 4.2 \cdot 10^{-6}\right):\\ \;\;\;\;\left(0.918938533204673 + x \cdot y\right) - x\\ \mathbf{else}:\\ \;\;\;\;x \cdot y + \left(0.918938533204673 - y \cdot 0.5\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (or (<= x -4.5) (not (<= x 4.2e-6)))
   (- (+ 0.918938533204673 (* x y)) x)
   (+ (* x y) (- 0.918938533204673 (* y 0.5)))))

C

double code(double x, double y) {
	double tmp;
	if ((x <= -4.5) || !(x <= 4.2e-6)) {
		tmp = (0.918938533204673 + (x * y)) - x;
	} else {
		tmp = (x * y) + (0.918938533204673 - (y * 0.5));
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if ((x <= (-4.5d0)) .or. (.not. (x <= 4.2d-6))) then
        tmp = (0.918938533204673d0 + (x * y)) - x
    else
        tmp = (x * y) + (0.918938533204673d0 - (y * 0.5d0))
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if ((x <= -4.5) || !(x <= 4.2e-6)) {
		tmp = (0.918938533204673 + (x * y)) - x;
	} else {
		tmp = (x * y) + (0.918938533204673 - (y * 0.5));
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if (x <= -4.5) or not (x <= 4.2e-6):
		tmp = (0.918938533204673 + (x * y)) - x
	else:
		tmp = (x * y) + (0.918938533204673 - (y * 0.5))
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if ((x <= -4.5) || !(x <= 4.2e-6))
		tmp = Float64(Float64(0.918938533204673 + Float64(x * y)) - x);
	else
		tmp = Float64(Float64(x * y) + Float64(0.918938533204673 - Float64(y * 0.5)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if ((x <= -4.5) || ~((x <= 4.2e-6)))
		tmp = (0.918938533204673 + (x * y)) - x;
	else
		tmp = (x * y) + (0.918938533204673 - (y * 0.5));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[Or[LessEqual[x, -4.5], N[Not[LessEqual[x, 4.2e-6]], $MachinePrecision]], N[(N[(0.918938533204673 + N[(x * y), $MachinePrecision]), $MachinePrecision] - x), $MachinePrecision], N[(N[(x * y), $MachinePrecision] + N[(0.918938533204673 - N[(y * 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -4.5 or 4.1999999999999996e-6 < x

   1. Initial program 100.0%

      \[\left(x \cdot \left(y - 1\right) - y \cdot 0.5\right) + 0.918938533204673 \]
   2. Step-by-step derivation

      1. cancel-sign-sub-inv 100.0%

         \[\leadsto \color{blue}{\left(x \cdot \left(y - 1\right) + \left(-y\right) \cdot 0.5\right)} + 0.918938533204673 \]

      2. +-commutative 100.0%

         \[\leadsto \color{blue}{\left(\left(-y\right) \cdot 0.5 + x \cdot \left(y - 1\right)\right)} + 0.918938533204673 \]

      3. distribute-lft-out-- 100.0%

         \[\leadsto \left(\left(-y\right) \cdot 0.5 + \color{blue}{\left(x \cdot y - x \cdot 1\right)}\right) + 0.918938533204673 \]

      4. cancel-sign-sub-inv 100.0%

         \[\leadsto \left(\left(-y\right) \cdot 0.5 + \color{blue}{\left(x \cdot y + \left(-x\right) \cdot 1\right)}\right) + 0.918938533204673 \]

      5. *-rgt-identity 100.0%

         \[\leadsto \left(\left(-y\right) \cdot 0.5 + \left(x \cdot y + \color{blue}{\left(-x\right)}\right)\right) + 0.918938533204673 \]

      6. associate-+r+ 100.0%

         \[\leadsto \color{blue}{\left(\left(\left(-y\right) \cdot 0.5 + x \cdot y\right) + \left(-x\right)\right)} + 0.918938533204673 \]

      7. associate-+l+ 100.0%

         \[\leadsto \color{blue}{\left(\left(-y\right) \cdot 0.5 + x \cdot y\right) + \left(\left(-x\right) + 0.918938533204673\right)} \]

      8. +-commutative 100.0%

         \[\leadsto \left(\left(-y\right) \cdot 0.5 + x \cdot y\right) + \color{blue}{\left(0.918938533204673 + \left(-x\right)\right)} \]

      9. unsub-neg 100.0%

         \[\leadsto \left(\left(-y\right) \cdot 0.5 + x \cdot y\right) + \color{blue}{\left(0.918938533204673 - x\right)} \]

      10. associate-+r- 100.0%

          \[\leadsto \color{blue}{\left(\left(\left(-y\right) \cdot 0.5 + x \cdot y\right) + 0.918938533204673\right) - x} \]

      11. *-commutative 100.0%

          \[\leadsto \left(\left(\left(-y\right) \cdot 0.5 + \color{blue}{y \cdot x}\right) + 0.918938533204673\right) - x \]

      12. distribute-lft-neg-out 100.0%

          \[\leadsto \left(\left(\color{blue}{\left(-y \cdot 0.5\right)} + y \cdot x\right) + 0.918938533204673\right) - x \]

      13. distribute-rgt-neg-in 100.0%

          \[\leadsto \left(\left(\color{blue}{y \cdot \left(-0.5\right)} + y \cdot x\right) + 0.918938533204673\right) - x \]

      14. distribute-lft-out 100.0%

          \[\leadsto \left(\color{blue}{y \cdot \left(\left(-0.5\right) + x\right)} + 0.918938533204673\right) - x \]

      15. fma-def 100.0%

          \[\leadsto \color{blue}{\mathsf{fma}\left(y, \left(-0.5\right) + x, 0.918938533204673\right)} - x \]

      16. +-commutative 100.0%

          \[\leadsto \mathsf{fma}\left(y, \color{blue}{x + \left(-0.5\right)}, 0.918938533204673\right) - x \]

      17. sub-neg 100.0%

          \[\leadsto \mathsf{fma}\left(y, \color{blue}{x - 0.5}, 0.918938533204673\right) - x \]

   3. Simplified 100.0%

      \[\leadsto \color{blue}{\mathsf{fma}\left(y, x - 0.5, 0.918938533204673\right) - x} \]
   4. Step-by-step derivation

      1. fma-udef 100.0%

         \[\leadsto \color{blue}{\left(y \cdot \left(x - 0.5\right) + 0.918938533204673\right)} - x \]

      2. *-un-lft-identity 100.0%

         \[\leadsto \left(y \cdot \color{blue}{\left(1 \cdot \left(x - 0.5\right)\right)} + 0.918938533204673\right) - x \]

      3. *-un-lft-identity 100.0%

         \[\leadsto \left(y \cdot \color{blue}{\left(x - 0.5\right)} + 0.918938533204673\right) - x \]

      4. sub-neg 100.0%

         \[\leadsto \left(y \cdot \color{blue}{\left(x + \left(-0.5\right)\right)} + 0.918938533204673\right) - x \]

      5. metadata-eval 100.0%

         \[\leadsto \left(y \cdot \left(x + \color{blue}{-0.5}\right) + 0.918938533204673\right) - x \]

   5. Applied egg-rr 100.0%

      \[\leadsto \color{blue}{\left(y \cdot \left(x + -0.5\right) + 0.918938533204673\right)} - x \]

   6. Taylor expanded in x around inf 98.8%

      \[\leadsto \left(\color{blue}{y \cdot x} + 0.918938533204673\right) - x \]

   if -4.5 < x < 4.1999999999999996e-6

   1. Initial program 100.0%

      \[\left(x \cdot \left(y - 1\right) - y \cdot 0.5\right) + 0.918938533204673 \]
   2. Step-by-step derivation

      1. associate-+l- 100.0%

         \[\leadsto \color{blue}{x \cdot \left(y - 1\right) - \left(y \cdot 0.5 - 0.918938533204673\right)} \]

      2. sub-neg 100.0%

         \[\leadsto x \cdot \color{blue}{\left(y + \left(-1\right)\right)} - \left(y \cdot 0.5 - 0.918938533204673\right) \]

      3. metadata-eval 100.0%

         \[\leadsto x \cdot \left(y + \color{blue}{-1}\right) - \left(y \cdot 0.5 - 0.918938533204673\right) \]

   3. Simplified 100.0%

      \[\leadsto \color{blue}{x \cdot \left(y + -1\right) - \left(y \cdot 0.5 - 0.918938533204673\right)} \]

   4. Taylor expanded in y around inf 99.9%

      \[\leadsto \color{blue}{y \cdot x} - \left(y \cdot 0.5 - 0.918938533204673\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 99.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -4.5 \lor \neg \left(x \leq 4.2 \cdot 10^{-6}\right):\\ \;\;\;\;\left(0.918938533204673 + x \cdot y\right) - x\\ \mathbf{else}:\\ \;\;\;\;x \cdot y + \left(0.918938533204673 - y \cdot 0.5\right)\\ \end{array} \]

Alternative 4: 98.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -5200000 \lor \neg \left(x \leq 950\right):\\ \;\;\;\;x \cdot y - x\\ \mathbf{else}:\\ \;\;\;\;\left(0.918938533204673 + y \cdot -0.5\right) - x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (or (<= x -5200000.0) (not (<= x 950.0)))
   (- (* x y) x)
   (- (+ 0.918938533204673 (* y -0.5)) x)))

C

double code(double x, double y) {
	double tmp;
	if ((x <= -5200000.0) || !(x <= 950.0)) {
		tmp = (x * y) - x;
	} else {
		tmp = (0.918938533204673 + (y * -0.5)) - x;
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if ((x <= (-5200000.0d0)) .or. (.not. (x <= 950.0d0))) then
        tmp = (x * y) - x
    else
        tmp = (0.918938533204673d0 + (y * (-0.5d0))) - x
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if ((x <= -5200000.0) || !(x <= 950.0)) {
		tmp = (x * y) - x;
	} else {
		tmp = (0.918938533204673 + (y * -0.5)) - x;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if (x <= -5200000.0) or not (x <= 950.0):
		tmp = (x * y) - x
	else:
		tmp = (0.918938533204673 + (y * -0.5)) - x
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if ((x <= -5200000.0) || !(x <= 950.0))
		tmp = Float64(Float64(x * y) - x);
	else
		tmp = Float64(Float64(0.918938533204673 + Float64(y * -0.5)) - x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if ((x <= -5200000.0) || ~((x <= 950.0)))
		tmp = (x * y) - x;
	else
		tmp = (0.918938533204673 + (y * -0.5)) - x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[Or[LessEqual[x, -5200000.0], N[Not[LessEqual[x, 950.0]], $MachinePrecision]], N[(N[(x * y), $MachinePrecision] - x), $MachinePrecision], N[(N[(0.918938533204673 + N[(y * -0.5), $MachinePrecision]), $MachinePrecision] - x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -5.2e6 or 950 < x

   1. Initial program 100.0%

      \[\left(x \cdot \left(y - 1\right) - y \cdot 0.5\right) + 0.918938533204673 \]
   2. Step-by-step derivation

      1. cancel-sign-sub-inv 100.0%

         \[\leadsto \color{blue}{\left(x \cdot \left(y - 1\right) + \left(-y\right) \cdot 0.5\right)} + 0.918938533204673 \]

      2. +-commutative 100.0%

         \[\leadsto \color{blue}{\left(\left(-y\right) \cdot 0.5 + x \cdot \left(y - 1\right)\right)} + 0.918938533204673 \]

      3. distribute-lft-out-- 100.0%

         \[\leadsto \left(\left(-y\right) \cdot 0.5 + \color{blue}{\left(x \cdot y - x \cdot 1\right)}\right) + 0.918938533204673 \]

      4. cancel-sign-sub-inv 100.0%

         \[\leadsto \left(\left(-y\right) \cdot 0.5 + \color{blue}{\left(x \cdot y + \left(-x\right) \cdot 1\right)}\right) + 0.918938533204673 \]

      5. *-rgt-identity 100.0%

         \[\leadsto \left(\left(-y\right) \cdot 0.5 + \left(x \cdot y + \color{blue}{\left(-x\right)}\right)\right) + 0.918938533204673 \]

      6. associate-+r+ 100.0%

         \[\leadsto \color{blue}{\left(\left(\left(-y\right) \cdot 0.5 + x \cdot y\right) + \left(-x\right)\right)} + 0.918938533204673 \]

      7. associate-+l+ 100.0%

         \[\leadsto \color{blue}{\left(\left(-y\right) \cdot 0.5 + x \cdot y\right) + \left(\left(-x\right) + 0.918938533204673\right)} \]

      8. +-commutative 100.0%

         \[\leadsto \left(\left(-y\right) \cdot 0.5 + x \cdot y\right) + \color{blue}{\left(0.918938533204673 + \left(-x\right)\right)} \]

      9. unsub-neg 100.0%

         \[\leadsto \left(\left(-y\right) \cdot 0.5 + x \cdot y\right) + \color{blue}{\left(0.918938533204673 - x\right)} \]

      10. associate-+r- 100.0%

          \[\leadsto \color{blue}{\left(\left(\left(-y\right) \cdot 0.5 + x \cdot y\right) + 0.918938533204673\right) - x} \]

      11. *-commutative 100.0%

          \[\leadsto \left(\left(\left(-y\right) \cdot 0.5 + \color{blue}{y \cdot x}\right) + 0.918938533204673\right) - x \]

      12. distribute-lft-neg-out 100.0%

          \[\leadsto \left(\left(\color{blue}{\left(-y \cdot 0.5\right)} + y \cdot x\right) + 0.918938533204673\right) - x \]

      13. distribute-rgt-neg-in 100.0%

          \[\leadsto \left(\left(\color{blue}{y \cdot \left(-0.5\right)} + y \cdot x\right) + 0.918938533204673\right) - x \]

      14. distribute-lft-out 100.0%

          \[\leadsto \left(\color{blue}{y \cdot \left(\left(-0.5\right) + x\right)} + 0.918938533204673\right) - x \]

      15. fma-def 100.0%

          \[\leadsto \color{blue}{\mathsf{fma}\left(y, \left(-0.5\right) + x, 0.918938533204673\right)} - x \]

      16. +-commutative 100.0%

          \[\leadsto \mathsf{fma}\left(y, \color{blue}{x + \left(-0.5\right)}, 0.918938533204673\right) - x \]

      17. sub-neg 100.0%

          \[\leadsto \mathsf{fma}\left(y, \color{blue}{x - 0.5}, 0.918938533204673\right) - x \]

   3. Simplified 100.0%

      \[\leadsto \color{blue}{\mathsf{fma}\left(y, x - 0.5, 0.918938533204673\right) - x} \]
   4. Step-by-step derivation

      1. fma-udef 100.0%

         \[\leadsto \color{blue}{\left(y \cdot \left(x - 0.5\right) + 0.918938533204673\right)} - x \]

      2. *-un-lft-identity 100.0%

         \[\leadsto \left(y \cdot \color{blue}{\left(1 \cdot \left(x - 0.5\right)\right)} + 0.918938533204673\right) - x \]

      3. *-un-lft-identity 100.0%

         \[\leadsto \left(y \cdot \color{blue}{\left(x - 0.5\right)} + 0.918938533204673\right) - x \]

      4. sub-neg 100.0%

         \[\leadsto \left(y \cdot \color{blue}{\left(x + \left(-0.5\right)\right)} + 0.918938533204673\right) - x \]

      5. metadata-eval 100.0%

         \[\leadsto \left(y \cdot \left(x + \color{blue}{-0.5}\right) + 0.918938533204673\right) - x \]

   5. Applied egg-rr 100.0%

      \[\leadsto \color{blue}{\left(y \cdot \left(x + -0.5\right) + 0.918938533204673\right)} - x \]

   6. Taylor expanded in x around inf 99.5%

      \[\leadsto \color{blue}{\left(y - 1\right) \cdot x} \]
   7. Step-by-step derivation

      1. *-commutative 99.5%

         \[\leadsto \color{blue}{x \cdot \left(y - 1\right)} \]

      2. sub-neg 99.5%

         \[\leadsto x \cdot \color{blue}{\left(y + \left(-1\right)\right)} \]

      3. metadata-eval 99.5%

         \[\leadsto x \cdot \left(y + \color{blue}{-1}\right) \]

      4. distribute-rgt-in 99.5%

         \[\leadsto \color{blue}{y \cdot x + -1 \cdot x} \]

      5. neg-mul-1 99.5%

         \[\leadsto y \cdot x + \color{blue}{\left(-x\right)} \]

      6. sub-neg 99.5%

         \[\leadsto \color{blue}{y \cdot x - x} \]

   8. Simplified 99.5%

      \[\leadsto \color{blue}{y \cdot x - x} \]

   if -5.2e6 < x < 950

   1. Initial program 100.0%

      \[\left(x \cdot \left(y - 1\right) - y \cdot 0.5\right) + 0.918938533204673 \]
   2. Step-by-step derivation

      1. cancel-sign-sub-inv 100.0%

         \[\leadsto \color{blue}{\left(x \cdot \left(y - 1\right) + \left(-y\right) \cdot 0.5\right)} + 0.918938533204673 \]

      2. +-commutative 100.0%

         \[\leadsto \color{blue}{\left(\left(-y\right) \cdot 0.5 + x \cdot \left(y - 1\right)\right)} + 0.918938533204673 \]

      3. distribute-lft-out-- 100.0%

         \[\leadsto \left(\left(-y\right) \cdot 0.5 + \color{blue}{\left(x \cdot y - x \cdot 1\right)}\right) + 0.918938533204673 \]

      4. cancel-sign-sub-inv 100.0%

         \[\leadsto \left(\left(-y\right) \cdot 0.5 + \color{blue}{\left(x \cdot y + \left(-x\right) \cdot 1\right)}\right) + 0.918938533204673 \]

      5. *-rgt-identity 100.0%

         \[\leadsto \left(\left(-y\right) \cdot 0.5 + \left(x \cdot y + \color{blue}{\left(-x\right)}\right)\right) + 0.918938533204673 \]

      6. associate-+r+ 100.0%

         \[\leadsto \color{blue}{\left(\left(\left(-y\right) \cdot 0.5 + x \cdot y\right) + \left(-x\right)\right)} + 0.918938533204673 \]

      7. associate-+l+ 100.0%

         \[\leadsto \color{blue}{\left(\left(-y\right) \cdot 0.5 + x \cdot y\right) + \left(\left(-x\right) + 0.918938533204673\right)} \]

      8. +-commutative 100.0%

         \[\leadsto \left(\left(-y\right) \cdot 0.5 + x \cdot y\right) + \color{blue}{\left(0.918938533204673 + \left(-x\right)\right)} \]

      9. unsub-neg 100.0%

         \[\leadsto \left(\left(-y\right) \cdot 0.5 + x \cdot y\right) + \color{blue}{\left(0.918938533204673 - x\right)} \]

      10. associate-+r- 100.0%

          \[\leadsto \color{blue}{\left(\left(\left(-y\right) \cdot 0.5 + x \cdot y\right) + 0.918938533204673\right) - x} \]

      11. *-commutative 100.0%

          \[\leadsto \left(\left(\left(-y\right) \cdot 0.5 + \color{blue}{y \cdot x}\right) + 0.918938533204673\right) - x \]

      12. distribute-lft-neg-out 100.0%

          \[\leadsto \left(\left(\color{blue}{\left(-y \cdot 0.5\right)} + y \cdot x\right) + 0.918938533204673\right) - x \]

      13. distribute-rgt-neg-in 100.0%

          \[\leadsto \left(\left(\color{blue}{y \cdot \left(-0.5\right)} + y \cdot x\right) + 0.918938533204673\right) - x \]

      14. distribute-lft-out 99.9%

          \[\leadsto \left(\color{blue}{y \cdot \left(\left(-0.5\right) + x\right)} + 0.918938533204673\right) - x \]

      15. fma-def 99.9%

          \[\leadsto \color{blue}{\mathsf{fma}\left(y, \left(-0.5\right) + x, 0.918938533204673\right)} - x \]

      16. +-commutative 99.9%

          \[\leadsto \mathsf{fma}\left(y, \color{blue}{x + \left(-0.5\right)}, 0.918938533204673\right) - x \]

      17. sub-neg 99.9%

          \[\leadsto \mathsf{fma}\left(y, \color{blue}{x - 0.5}, 0.918938533204673\right) - x \]

   3. Simplified 99.9%

      \[\leadsto \color{blue}{\mathsf{fma}\left(y, x - 0.5, 0.918938533204673\right) - x} \]

   4. Taylor expanded in x around 0 98.2%

      \[\leadsto \color{blue}{\left(-0.5 \cdot y + 0.918938533204673\right)} - x \]
3. Recombined 2 regimes into one program.

4. Final simplification 98.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -5200000 \lor \neg \left(x \leq 950\right):\\ \;\;\;\;x \cdot y - x\\ \mathbf{else}:\\ \;\;\;\;\left(0.918938533204673 + y \cdot -0.5\right) - x\\ \end{array} \]

Alternative 5: 98.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -19000:\\ \;\;\;\;\left(0.918938533204673 + x \cdot y\right) - x\\ \mathbf{elif}\;x \leq 260:\\ \;\;\;\;\left(0.918938533204673 + y \cdot -0.5\right) - x\\ \mathbf{else}:\\ \;\;\;\;x \cdot y - x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x -19000.0)
   (- (+ 0.918938533204673 (* x y)) x)
   (if (<= x 260.0) (- (+ 0.918938533204673 (* y -0.5)) x) (- (* x y) x))))

C

double code(double x, double y) {
	double tmp;
	if (x <= -19000.0) {
		tmp = (0.918938533204673 + (x * y)) - x;
	} else if (x <= 260.0) {
		tmp = (0.918938533204673 + (y * -0.5)) - x;
	} else {
		tmp = (x * y) - x;
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (x <= (-19000.0d0)) then
        tmp = (0.918938533204673d0 + (x * y)) - x
    else if (x <= 260.0d0) then
        tmp = (0.918938533204673d0 + (y * (-0.5d0))) - x
    else
        tmp = (x * y) - x
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (x <= -19000.0) {
		tmp = (0.918938533204673 + (x * y)) - x;
	} else if (x <= 260.0) {
		tmp = (0.918938533204673 + (y * -0.5)) - x;
	} else {
		tmp = (x * y) - x;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if x <= -19000.0:
		tmp = (0.918938533204673 + (x * y)) - x
	elif x <= 260.0:
		tmp = (0.918938533204673 + (y * -0.5)) - x
	else:
		tmp = (x * y) - x
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (x <= -19000.0)
		tmp = Float64(Float64(0.918938533204673 + Float64(x * y)) - x);
	elseif (x <= 260.0)
		tmp = Float64(Float64(0.918938533204673 + Float64(y * -0.5)) - x);
	else
		tmp = Float64(Float64(x * y) - x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= -19000.0)
		tmp = (0.918938533204673 + (x * y)) - x;
	elseif (x <= 260.0)
		tmp = (0.918938533204673 + (y * -0.5)) - x;
	else
		tmp = (x * y) - x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[x, -19000.0], N[(N[(0.918938533204673 + N[(x * y), $MachinePrecision]), $MachinePrecision] - x), $MachinePrecision], If[LessEqual[x, 260.0], N[(N[(0.918938533204673 + N[(y * -0.5), $MachinePrecision]), $MachinePrecision] - x), $MachinePrecision], N[(N[(x * y), $MachinePrecision] - x), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -19000

   1. Initial program 100.0%

      \[\left(x \cdot \left(y - 1\right) - y \cdot 0.5\right) + 0.918938533204673 \]
   2. Step-by-step derivation

      1. cancel-sign-sub-inv 100.0%

         \[\leadsto \color{blue}{\left(x \cdot \left(y - 1\right) + \left(-y\right) \cdot 0.5\right)} + 0.918938533204673 \]

      2. +-commutative 100.0%

         \[\leadsto \color{blue}{\left(\left(-y\right) \cdot 0.5 + x \cdot \left(y - 1\right)\right)} + 0.918938533204673 \]

      3. distribute-lft-out-- 100.0%

         \[\leadsto \left(\left(-y\right) \cdot 0.5 + \color{blue}{\left(x \cdot y - x \cdot 1\right)}\right) + 0.918938533204673 \]

      4. cancel-sign-sub-inv 100.0%

         \[\leadsto \left(\left(-y\right) \cdot 0.5 + \color{blue}{\left(x \cdot y + \left(-x\right) \cdot 1\right)}\right) + 0.918938533204673 \]

      5. *-rgt-identity 100.0%

         \[\leadsto \left(\left(-y\right) \cdot 0.5 + \left(x \cdot y + \color{blue}{\left(-x\right)}\right)\right) + 0.918938533204673 \]

      6. associate-+r+ 100.0%

         \[\leadsto \color{blue}{\left(\left(\left(-y\right) \cdot 0.5 + x \cdot y\right) + \left(-x\right)\right)} + 0.918938533204673 \]

      7. associate-+l+ 100.0%

         \[\leadsto \color{blue}{\left(\left(-y\right) \cdot 0.5 + x \cdot y\right) + \left(\left(-x\right) + 0.918938533204673\right)} \]

      8. +-commutative 100.0%

         \[\leadsto \left(\left(-y\right) \cdot 0.5 + x \cdot y\right) + \color{blue}{\left(0.918938533204673 + \left(-x\right)\right)} \]

      9. unsub-neg 100.0%

         \[\leadsto \left(\left(-y\right) \cdot 0.5 + x \cdot y\right) + \color{blue}{\left(0.918938533204673 - x\right)} \]

      10. associate-+r- 100.0%

          \[\leadsto \color{blue}{\left(\left(\left(-y\right) \cdot 0.5 + x \cdot y\right) + 0.918938533204673\right) - x} \]

      11. *-commutative 100.0%

          \[\leadsto \left(\left(\left(-y\right) \cdot 0.5 + \color{blue}{y \cdot x}\right) + 0.918938533204673\right) - x \]

      12. distribute-lft-neg-out 100.0%

          \[\leadsto \left(\left(\color{blue}{\left(-y \cdot 0.5\right)} + y \cdot x\right) + 0.918938533204673\right) - x \]

      13. distribute-rgt-neg-in 100.0%

          \[\leadsto \left(\left(\color{blue}{y \cdot \left(-0.5\right)} + y \cdot x\right) + 0.918938533204673\right) - x \]

      14. distribute-lft-out 100.0%

          \[\leadsto \left(\color{blue}{y \cdot \left(\left(-0.5\right) + x\right)} + 0.918938533204673\right) - x \]

      15. fma-def 100.0%

          \[\leadsto \color{blue}{\mathsf{fma}\left(y, \left(-0.5\right) + x, 0.918938533204673\right)} - x \]

      16. +-commutative 100.0%

          \[\leadsto \mathsf{fma}\left(y, \color{blue}{x + \left(-0.5\right)}, 0.918938533204673\right) - x \]

      17. sub-neg 100.0%

          \[\leadsto \mathsf{fma}\left(y, \color{blue}{x - 0.5}, 0.918938533204673\right) - x \]

   3. Simplified 100.0%

      \[\leadsto \color{blue}{\mathsf{fma}\left(y, x - 0.5, 0.918938533204673\right) - x} \]
   4. Step-by-step derivation

      1. fma-udef 100.0%

         \[\leadsto \color{blue}{\left(y \cdot \left(x - 0.5\right) + 0.918938533204673\right)} - x \]

      2. *-un-lft-identity 100.0%

         \[\leadsto \left(y \cdot \color{blue}{\left(1 \cdot \left(x - 0.5\right)\right)} + 0.918938533204673\right) - x \]

      3. *-un-lft-identity 100.0%

         \[\leadsto \left(y \cdot \color{blue}{\left(x - 0.5\right)} + 0.918938533204673\right) - x \]

      4. sub-neg 100.0%

         \[\leadsto \left(y \cdot \color{blue}{\left(x + \left(-0.5\right)\right)} + 0.918938533204673\right) - x \]

      5. metadata-eval 100.0%

         \[\leadsto \left(y \cdot \left(x + \color{blue}{-0.5}\right) + 0.918938533204673\right) - x \]

   5. Applied egg-rr 100.0%

      \[\leadsto \color{blue}{\left(y \cdot \left(x + -0.5\right) + 0.918938533204673\right)} - x \]

   6. Taylor expanded in x around inf 100.0%

      \[\leadsto \left(\color{blue}{y \cdot x} + 0.918938533204673\right) - x \]

   if -19000 < x < 260

   1. Initial program 100.0%

      \[\left(x \cdot \left(y - 1\right) - y \cdot 0.5\right) + 0.918938533204673 \]
   2. Step-by-step derivation

      1. cancel-sign-sub-inv 100.0%

         \[\leadsto \color{blue}{\left(x \cdot \left(y - 1\right) + \left(-y\right) \cdot 0.5\right)} + 0.918938533204673 \]

      2. +-commutative 100.0%

         \[\leadsto \color{blue}{\left(\left(-y\right) \cdot 0.5 + x \cdot \left(y - 1\right)\right)} + 0.918938533204673 \]

      3. distribute-lft-out-- 100.0%

         \[\leadsto \left(\left(-y\right) \cdot 0.5 + \color{blue}{\left(x \cdot y - x \cdot 1\right)}\right) + 0.918938533204673 \]

      4. cancel-sign-sub-inv 100.0%

         \[\leadsto \left(\left(-y\right) \cdot 0.5 + \color{blue}{\left(x \cdot y + \left(-x\right) \cdot 1\right)}\right) + 0.918938533204673 \]

      5. *-rgt-identity 100.0%

         \[\leadsto \left(\left(-y\right) \cdot 0.5 + \left(x \cdot y + \color{blue}{\left(-x\right)}\right)\right) + 0.918938533204673 \]

      6. associate-+r+ 100.0%

         \[\leadsto \color{blue}{\left(\left(\left(-y\right) \cdot 0.5 + x \cdot y\right) + \left(-x\right)\right)} + 0.918938533204673 \]

      7. associate-+l+ 100.0%

         \[\leadsto \color{blue}{\left(\left(-y\right) \cdot 0.5 + x \cdot y\right) + \left(\left(-x\right) + 0.918938533204673\right)} \]

      8. +-commutative 100.0%

         \[\leadsto \left(\left(-y\right) \cdot 0.5 + x \cdot y\right) + \color{blue}{\left(0.918938533204673 + \left(-x\right)\right)} \]

      9. unsub-neg 100.0%

         \[\leadsto \left(\left(-y\right) \cdot 0.5 + x \cdot y\right) + \color{blue}{\left(0.918938533204673 - x\right)} \]

      10. associate-+r- 100.0%

          \[\leadsto \color{blue}{\left(\left(\left(-y\right) \cdot 0.5 + x \cdot y\right) + 0.918938533204673\right) - x} \]

      11. *-commutative 100.0%

          \[\leadsto \left(\left(\left(-y\right) \cdot 0.5 + \color{blue}{y \cdot x}\right) + 0.918938533204673\right) - x \]

      12. distribute-lft-neg-out 100.0%

          \[\leadsto \left(\left(\color{blue}{\left(-y \cdot 0.5\right)} + y \cdot x\right) + 0.918938533204673\right) - x \]

      13. distribute-rgt-neg-in 100.0%

          \[\leadsto \left(\left(\color{blue}{y \cdot \left(-0.5\right)} + y \cdot x\right) + 0.918938533204673\right) - x \]

      14. distribute-lft-out 99.9%

          \[\leadsto \left(\color{blue}{y \cdot \left(\left(-0.5\right) + x\right)} + 0.918938533204673\right) - x \]

      15. fma-def 99.9%

          \[\leadsto \color{blue}{\mathsf{fma}\left(y, \left(-0.5\right) + x, 0.918938533204673\right)} - x \]

      16. +-commutative 99.9%

          \[\leadsto \mathsf{fma}\left(y, \color{blue}{x + \left(-0.5\right)}, 0.918938533204673\right) - x \]

      17. sub-neg 99.9%

          \[\leadsto \mathsf{fma}\left(y, \color{blue}{x - 0.5}, 0.918938533204673\right) - x \]

   3. Simplified 99.9%

      \[\leadsto \color{blue}{\mathsf{fma}\left(y, x - 0.5, 0.918938533204673\right) - x} \]

   4. Taylor expanded in x around 0 98.4%

      \[\leadsto \color{blue}{\left(-0.5 \cdot y + 0.918938533204673\right)} - x \]

   if 260 < x

   1. Initial program 99.9%

      \[\left(x \cdot \left(y - 1\right) - y \cdot 0.5\right) + 0.918938533204673 \]
   2. Step-by-step derivation

      1. cancel-sign-sub-inv 99.9%

         \[\leadsto \color{blue}{\left(x \cdot \left(y - 1\right) + \left(-y\right) \cdot 0.5\right)} + 0.918938533204673 \]

      2. +-commutative 99.9%

         \[\leadsto \color{blue}{\left(\left(-y\right) \cdot 0.5 + x \cdot \left(y - 1\right)\right)} + 0.918938533204673 \]

      3. distribute-lft-out-- 100.0%

         \[\leadsto \left(\left(-y\right) \cdot 0.5 + \color{blue}{\left(x \cdot y - x \cdot 1\right)}\right) + 0.918938533204673 \]

      4. cancel-sign-sub-inv 100.0%

         \[\leadsto \left(\left(-y\right) \cdot 0.5 + \color{blue}{\left(x \cdot y + \left(-x\right) \cdot 1\right)}\right) + 0.918938533204673 \]

      5. *-rgt-identity 100.0%

         \[\leadsto \left(\left(-y\right) \cdot 0.5 + \left(x \cdot y + \color{blue}{\left(-x\right)}\right)\right) + 0.918938533204673 \]

      6. associate-+r+ 100.0%

         \[\leadsto \color{blue}{\left(\left(\left(-y\right) \cdot 0.5 + x \cdot y\right) + \left(-x\right)\right)} + 0.918938533204673 \]

      7. associate-+l+ 100.0%

         \[\leadsto \color{blue}{\left(\left(-y\right) \cdot 0.5 + x \cdot y\right) + \left(\left(-x\right) + 0.918938533204673\right)} \]

      8. +-commutative 100.0%

         \[\leadsto \left(\left(-y\right) \cdot 0.5 + x \cdot y\right) + \color{blue}{\left(0.918938533204673 + \left(-x\right)\right)} \]

      9. unsub-neg 100.0%

         \[\leadsto \left(\left(-y\right) \cdot 0.5 + x \cdot y\right) + \color{blue}{\left(0.918938533204673 - x\right)} \]

      10. associate-+r- 100.0%

          \[\leadsto \color{blue}{\left(\left(\left(-y\right) \cdot 0.5 + x \cdot y\right) + 0.918938533204673\right) - x} \]

      11. *-commutative 100.0%

          \[\leadsto \left(\left(\left(-y\right) \cdot 0.5 + \color{blue}{y \cdot x}\right) + 0.918938533204673\right) - x \]

      12. distribute-lft-neg-out 100.0%

          \[\leadsto \left(\left(\color{blue}{\left(-y \cdot 0.5\right)} + y \cdot x\right) + 0.918938533204673\right) - x \]

      13. distribute-rgt-neg-in 100.0%

          \[\leadsto \left(\left(\color{blue}{y \cdot \left(-0.5\right)} + y \cdot x\right) + 0.918938533204673\right) - x \]

      14. distribute-lft-out 100.0%

          \[\leadsto \left(\color{blue}{y \cdot \left(\left(-0.5\right) + x\right)} + 0.918938533204673\right) - x \]

      15. fma-def 100.0%

          \[\leadsto \color{blue}{\mathsf{fma}\left(y, \left(-0.5\right) + x, 0.918938533204673\right)} - x \]

      16. +-commutative 100.0%

          \[\leadsto \mathsf{fma}\left(y, \color{blue}{x + \left(-0.5\right)}, 0.918938533204673\right) - x \]

      17. sub-neg 100.0%

          \[\leadsto \mathsf{fma}\left(y, \color{blue}{x - 0.5}, 0.918938533204673\right) - x \]

   3. Simplified 100.0%

      \[\leadsto \color{blue}{\mathsf{fma}\left(y, x - 0.5, 0.918938533204673\right) - x} \]
   4. Step-by-step derivation

      1. fma-udef 100.0%

         \[\leadsto \color{blue}{\left(y \cdot \left(x - 0.5\right) + 0.918938533204673\right)} - x \]

      2. *-un-lft-identity 100.0%

         \[\leadsto \left(y \cdot \color{blue}{\left(1 \cdot \left(x - 0.5\right)\right)} + 0.918938533204673\right) - x \]

      3. *-un-lft-identity 100.0%

         \[\leadsto \left(y \cdot \color{blue}{\left(x - 0.5\right)} + 0.918938533204673\right) - x \]

      4. sub-neg 100.0%

         \[\leadsto \left(y \cdot \color{blue}{\left(x + \left(-0.5\right)\right)} + 0.918938533204673\right) - x \]

      5. metadata-eval 100.0%

         \[\leadsto \left(y \cdot \left(x + \color{blue}{-0.5}\right) + 0.918938533204673\right) - x \]

   5. Applied egg-rr 100.0%

      \[\leadsto \color{blue}{\left(y \cdot \left(x + -0.5\right) + 0.918938533204673\right)} - x \]

   6. Taylor expanded in x around inf 98.9%

      \[\leadsto \color{blue}{\left(y - 1\right) \cdot x} \]
   7. Step-by-step derivation

      1. *-commutative 98.9%

         \[\leadsto \color{blue}{x \cdot \left(y - 1\right)} \]

      2. sub-neg 98.9%

         \[\leadsto x \cdot \color{blue}{\left(y + \left(-1\right)\right)} \]

      3. metadata-eval 98.9%

         \[\leadsto x \cdot \left(y + \color{blue}{-1}\right) \]

      4. distribute-rgt-in 99.0%

         \[\leadsto \color{blue}{y \cdot x + -1 \cdot x} \]

      5. neg-mul-1 99.0%

         \[\leadsto y \cdot x + \color{blue}{\left(-x\right)} \]

      6. sub-neg 99.0%

         \[\leadsto \color{blue}{y \cdot x - x} \]

   8. Simplified 99.0%

      \[\leadsto \color{blue}{y \cdot x - x} \]
3. Recombined 3 regimes into one program.

4. Final simplification 99.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -19000:\\ \;\;\;\;\left(0.918938533204673 + x \cdot y\right) - x\\ \mathbf{elif}\;x \leq 260:\\ \;\;\;\;\left(0.918938533204673 + y \cdot -0.5\right) - x\\ \mathbf{else}:\\ \;\;\;\;x \cdot y - x\\ \end{array} \]

Alternative 6: 97.7% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -1.35 \lor \neg \left(y \leq 1.25\right):\\ \;\;\;\;y \cdot \left(x - 0.5\right)\\ \mathbf{else}:\\ \;\;\;\;0.918938533204673 - x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (or (<= y -1.35) (not (<= y 1.25)))
   (* y (- x 0.5))
   (- 0.918938533204673 x)))

C

double code(double x, double y) {
	double tmp;
	if ((y <= -1.35) || !(y <= 1.25)) {
		tmp = y * (x - 0.5);
	} else {
		tmp = 0.918938533204673 - x;
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if ((y <= (-1.35d0)) .or. (.not. (y <= 1.25d0))) then
        tmp = y * (x - 0.5d0)
    else
        tmp = 0.918938533204673d0 - x
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if ((y <= -1.35) || !(y <= 1.25)) {
		tmp = y * (x - 0.5);
	} else {
		tmp = 0.918938533204673 - x;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if (y <= -1.35) or not (y <= 1.25):
		tmp = y * (x - 0.5)
	else:
		tmp = 0.918938533204673 - x
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if ((y <= -1.35) || !(y <= 1.25))
		tmp = Float64(y * Float64(x - 0.5));
	else
		tmp = Float64(0.918938533204673 - x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if ((y <= -1.35) || ~((y <= 1.25)))
		tmp = y * (x - 0.5);
	else
		tmp = 0.918938533204673 - x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[Or[LessEqual[y, -1.35], N[Not[LessEqual[y, 1.25]], $MachinePrecision]], N[(y * N[(x - 0.5), $MachinePrecision]), $MachinePrecision], N[(0.918938533204673 - x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -1.3500000000000001 or 1.25 < y

   1. Initial program 100.0%

      \[\left(x \cdot \left(y - 1\right) - y \cdot 0.5\right) + 0.918938533204673 \]
   2. Step-by-step derivation

      1. associate-+l- 100.0%

         \[\leadsto \color{blue}{x \cdot \left(y - 1\right) - \left(y \cdot 0.5 - 0.918938533204673\right)} \]

      2. sub-neg 100.0%

         \[\leadsto x \cdot \color{blue}{\left(y + \left(-1\right)\right)} - \left(y \cdot 0.5 - 0.918938533204673\right) \]

      3. metadata-eval 100.0%

         \[\leadsto x \cdot \left(y + \color{blue}{-1}\right) - \left(y \cdot 0.5 - 0.918938533204673\right) \]

   3. Simplified 100.0%

      \[\leadsto \color{blue}{x \cdot \left(y + -1\right) - \left(y \cdot 0.5 - 0.918938533204673\right)} \]

   4. Taylor expanded in y around inf 97.2%

      \[\leadsto \color{blue}{y \cdot \left(x - 0.5\right)} \]

   if -1.3500000000000001 < y < 1.25

   1. Initial program 100.0%

      \[\left(x \cdot \left(y - 1\right) - y \cdot 0.5\right) + 0.918938533204673 \]
   2. Step-by-step derivation

      1. associate-+l- 100.0%

         \[\leadsto \color{blue}{x \cdot \left(y - 1\right) - \left(y \cdot 0.5 - 0.918938533204673\right)} \]

      2. sub-neg 100.0%

         \[\leadsto x \cdot \color{blue}{\left(y + \left(-1\right)\right)} - \left(y \cdot 0.5 - 0.918938533204673\right) \]

      3. metadata-eval 100.0%

         \[\leadsto x \cdot \left(y + \color{blue}{-1}\right) - \left(y \cdot 0.5 - 0.918938533204673\right) \]

   3. Simplified 100.0%

      \[\leadsto \color{blue}{x \cdot \left(y + -1\right) - \left(y \cdot 0.5 - 0.918938533204673\right)} \]

   4. Taylor expanded in y around 0 96.5%

      \[\leadsto \color{blue}{0.918938533204673 + -1 \cdot x} \]
   5. Step-by-step derivation

      1. neg-mul-1 96.5%

         \[\leadsto 0.918938533204673 + \color{blue}{\left(-x\right)} \]

      2. sub-neg 96.5%

         \[\leadsto \color{blue}{0.918938533204673 - x} \]

   6. Simplified 96.5%

      \[\leadsto \color{blue}{0.918938533204673 - x} \]
3. Recombined 2 regimes into one program.

4. Final simplification 96.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -1.35 \lor \neg \left(y \leq 1.25\right):\\ \;\;\;\;y \cdot \left(x - 0.5\right)\\ \mathbf{else}:\\ \;\;\;\;0.918938533204673 - x\\ \end{array} \]

Alternative 7: 97.9% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -0.72 \lor \neg \left(x \leq 0.56\right):\\ \;\;\;\;x \cdot \left(y + -1\right)\\ \mathbf{else}:\\ \;\;\;\;0.918938533204673 - y \cdot 0.5\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (or (<= x -0.72) (not (<= x 0.56)))
   (* x (+ y -1.0))
   (- 0.918938533204673 (* y 0.5))))

C

double code(double x, double y) {
	double tmp;
	if ((x <= -0.72) || !(x <= 0.56)) {
		tmp = x * (y + -1.0);
	} else {
		tmp = 0.918938533204673 - (y * 0.5);
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if ((x <= (-0.72d0)) .or. (.not. (x <= 0.56d0))) then
        tmp = x * (y + (-1.0d0))
    else
        tmp = 0.918938533204673d0 - (y * 0.5d0)
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if ((x <= -0.72) || !(x <= 0.56)) {
		tmp = x * (y + -1.0);
	} else {
		tmp = 0.918938533204673 - (y * 0.5);
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if (x <= -0.72) or not (x <= 0.56):
		tmp = x * (y + -1.0)
	else:
		tmp = 0.918938533204673 - (y * 0.5)
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if ((x <= -0.72) || !(x <= 0.56))
		tmp = Float64(x * Float64(y + -1.0));
	else
		tmp = Float64(0.918938533204673 - Float64(y * 0.5));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if ((x <= -0.72) || ~((x <= 0.56)))
		tmp = x * (y + -1.0);
	else
		tmp = 0.918938533204673 - (y * 0.5);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[Or[LessEqual[x, -0.72], N[Not[LessEqual[x, 0.56]], $MachinePrecision]], N[(x * N[(y + -1.0), $MachinePrecision]), $MachinePrecision], N[(0.918938533204673 - N[(y * 0.5), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -0.71999999999999997 or 0.56000000000000005 < x

   1. Initial program 100.0%

      \[\left(x \cdot \left(y - 1\right) - y \cdot 0.5\right) + 0.918938533204673 \]
   2. Step-by-step derivation

      1. cancel-sign-sub-inv 100.0%

         \[\leadsto \color{blue}{\left(x \cdot \left(y - 1\right) + \left(-y\right) \cdot 0.5\right)} + 0.918938533204673 \]

      2. +-commutative 100.0%

         \[\leadsto \color{blue}{\left(\left(-y\right) \cdot 0.5 + x \cdot \left(y - 1\right)\right)} + 0.918938533204673 \]

      3. distribute-lft-out-- 100.0%

         \[\leadsto \left(\left(-y\right) \cdot 0.5 + \color{blue}{\left(x \cdot y - x \cdot 1\right)}\right) + 0.918938533204673 \]

      4. cancel-sign-sub-inv 100.0%

         \[\leadsto \left(\left(-y\right) \cdot 0.5 + \color{blue}{\left(x \cdot y + \left(-x\right) \cdot 1\right)}\right) + 0.918938533204673 \]

      5. *-rgt-identity 100.0%

         \[\leadsto \left(\left(-y\right) \cdot 0.5 + \left(x \cdot y + \color{blue}{\left(-x\right)}\right)\right) + 0.918938533204673 \]

      6. associate-+r+ 100.0%

         \[\leadsto \color{blue}{\left(\left(\left(-y\right) \cdot 0.5 + x \cdot y\right) + \left(-x\right)\right)} + 0.918938533204673 \]

      7. associate-+l+ 100.0%

         \[\leadsto \color{blue}{\left(\left(-y\right) \cdot 0.5 + x \cdot y\right) + \left(\left(-x\right) + 0.918938533204673\right)} \]

      8. +-commutative 100.0%

         \[\leadsto \left(\left(-y\right) \cdot 0.5 + x \cdot y\right) + \color{blue}{\left(0.918938533204673 + \left(-x\right)\right)} \]

      9. unsub-neg 100.0%

         \[\leadsto \left(\left(-y\right) \cdot 0.5 + x \cdot y\right) + \color{blue}{\left(0.918938533204673 - x\right)} \]

      10. associate-+r- 100.0%

          \[\leadsto \color{blue}{\left(\left(\left(-y\right) \cdot 0.5 + x \cdot y\right) + 0.918938533204673\right) - x} \]

      11. *-commutative 100.0%

          \[\leadsto \left(\left(\left(-y\right) \cdot 0.5 + \color{blue}{y \cdot x}\right) + 0.918938533204673\right) - x \]

      12. distribute-lft-neg-out 100.0%

          \[\leadsto \left(\left(\color{blue}{\left(-y \cdot 0.5\right)} + y \cdot x\right) + 0.918938533204673\right) - x \]

      13. distribute-rgt-neg-in 100.0%

          \[\leadsto \left(\left(\color{blue}{y \cdot \left(-0.5\right)} + y \cdot x\right) + 0.918938533204673\right) - x \]

      14. distribute-lft-out 100.0%

          \[\leadsto \left(\color{blue}{y \cdot \left(\left(-0.5\right) + x\right)} + 0.918938533204673\right) - x \]

      15. fma-def 100.0%

          \[\leadsto \color{blue}{\mathsf{fma}\left(y, \left(-0.5\right) + x, 0.918938533204673\right)} - x \]

      16. +-commutative 100.0%

          \[\leadsto \mathsf{fma}\left(y, \color{blue}{x + \left(-0.5\right)}, 0.918938533204673\right) - x \]

      17. sub-neg 100.0%

          \[\leadsto \mathsf{fma}\left(y, \color{blue}{x - 0.5}, 0.918938533204673\right) - x \]

   3. Simplified 100.0%

      \[\leadsto \color{blue}{\mathsf{fma}\left(y, x - 0.5, 0.918938533204673\right) - x} \]
   4. Step-by-step derivation

      1. fma-udef 100.0%

         \[\leadsto \color{blue}{\left(y \cdot \left(x - 0.5\right) + 0.918938533204673\right)} - x \]

      2. *-un-lft-identity 100.0%

         \[\leadsto \left(y \cdot \color{blue}{\left(1 \cdot \left(x - 0.5\right)\right)} + 0.918938533204673\right) - x \]

      3. *-un-lft-identity 100.0%

         \[\leadsto \left(y \cdot \color{blue}{\left(x - 0.5\right)} + 0.918938533204673\right) - x \]

      4. sub-neg 100.0%

         \[\leadsto \left(y \cdot \color{blue}{\left(x + \left(-0.5\right)\right)} + 0.918938533204673\right) - x \]

      5. metadata-eval 100.0%

         \[\leadsto \left(y \cdot \left(x + \color{blue}{-0.5}\right) + 0.918938533204673\right) - x \]

   5. Applied egg-rr 100.0%

      \[\leadsto \color{blue}{\left(y \cdot \left(x + -0.5\right) + 0.918938533204673\right)} - x \]

   6. Taylor expanded in x around inf 98.6%

      \[\leadsto \color{blue}{\left(y - 1\right) \cdot x} \]

   if -0.71999999999999997 < x < 0.56000000000000005

   1. Initial program 100.0%

      \[\left(x \cdot \left(y - 1\right) - y \cdot 0.5\right) + 0.918938533204673 \]
   2. Step-by-step derivation

      1. associate-+l- 100.0%

         \[\leadsto \color{blue}{x \cdot \left(y - 1\right) - \left(y \cdot 0.5 - 0.918938533204673\right)} \]

      2. sub-neg 100.0%

         \[\leadsto x \cdot \color{blue}{\left(y + \left(-1\right)\right)} - \left(y \cdot 0.5 - 0.918938533204673\right) \]

      3. metadata-eval 100.0%

         \[\leadsto x \cdot \left(y + \color{blue}{-1}\right) - \left(y \cdot 0.5 - 0.918938533204673\right) \]

   3. Simplified 100.0%

      \[\leadsto \color{blue}{x \cdot \left(y + -1\right) - \left(y \cdot 0.5 - 0.918938533204673\right)} \]

   4. Taylor expanded in x around 0 96.9%

      \[\leadsto \color{blue}{0.918938533204673 - 0.5 \cdot y} \]
3. Recombined 2 regimes into one program.

4. Final simplification 97.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -0.72 \lor \neg \left(x \leq 0.56\right):\\ \;\;\;\;x \cdot \left(y + -1\right)\\ \mathbf{else}:\\ \;\;\;\;0.918938533204673 - y \cdot 0.5\\ \end{array} \]

Alternative 8: 97.9% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -0.76 \lor \neg \left(x \leq 0.75\right):\\ \;\;\;\;x \cdot y - x\\ \mathbf{else}:\\ \;\;\;\;0.918938533204673 - y \cdot 0.5\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (or (<= x -0.76) (not (<= x 0.75)))
   (- (* x y) x)
   (- 0.918938533204673 (* y 0.5))))

C

double code(double x, double y) {
	double tmp;
	if ((x <= -0.76) || !(x <= 0.75)) {
		tmp = (x * y) - x;
	} else {
		tmp = 0.918938533204673 - (y * 0.5);
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if ((x <= (-0.76d0)) .or. (.not. (x <= 0.75d0))) then
        tmp = (x * y) - x
    else
        tmp = 0.918938533204673d0 - (y * 0.5d0)
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if ((x <= -0.76) || !(x <= 0.75)) {
		tmp = (x * y) - x;
	} else {
		tmp = 0.918938533204673 - (y * 0.5);
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if (x <= -0.76) or not (x <= 0.75):
		tmp = (x * y) - x
	else:
		tmp = 0.918938533204673 - (y * 0.5)
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if ((x <= -0.76) || !(x <= 0.75))
		tmp = Float64(Float64(x * y) - x);
	else
		tmp = Float64(0.918938533204673 - Float64(y * 0.5));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if ((x <= -0.76) || ~((x <= 0.75)))
		tmp = (x * y) - x;
	else
		tmp = 0.918938533204673 - (y * 0.5);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[Or[LessEqual[x, -0.76], N[Not[LessEqual[x, 0.75]], $MachinePrecision]], N[(N[(x * y), $MachinePrecision] - x), $MachinePrecision], N[(0.918938533204673 - N[(y * 0.5), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -0.76000000000000001 or 0.75 < x

   1. Initial program 100.0%

      \[\left(x \cdot \left(y - 1\right) - y \cdot 0.5\right) + 0.918938533204673 \]
   2. Step-by-step derivation

      1. cancel-sign-sub-inv 100.0%

         \[\leadsto \color{blue}{\left(x \cdot \left(y - 1\right) + \left(-y\right) \cdot 0.5\right)} + 0.918938533204673 \]

      2. +-commutative 100.0%

         \[\leadsto \color{blue}{\left(\left(-y\right) \cdot 0.5 + x \cdot \left(y - 1\right)\right)} + 0.918938533204673 \]

      3. distribute-lft-out-- 100.0%

         \[\leadsto \left(\left(-y\right) \cdot 0.5 + \color{blue}{\left(x \cdot y - x \cdot 1\right)}\right) + 0.918938533204673 \]

      4. cancel-sign-sub-inv 100.0%

         \[\leadsto \left(\left(-y\right) \cdot 0.5 + \color{blue}{\left(x \cdot y + \left(-x\right) \cdot 1\right)}\right) + 0.918938533204673 \]

      5. *-rgt-identity 100.0%

         \[\leadsto \left(\left(-y\right) \cdot 0.5 + \left(x \cdot y + \color{blue}{\left(-x\right)}\right)\right) + 0.918938533204673 \]

      6. associate-+r+ 100.0%

         \[\leadsto \color{blue}{\left(\left(\left(-y\right) \cdot 0.5 + x \cdot y\right) + \left(-x\right)\right)} + 0.918938533204673 \]

      7. associate-+l+ 100.0%

         \[\leadsto \color{blue}{\left(\left(-y\right) \cdot 0.5 + x \cdot y\right) + \left(\left(-x\right) + 0.918938533204673\right)} \]

      8. +-commutative 100.0%

         \[\leadsto \left(\left(-y\right) \cdot 0.5 + x \cdot y\right) + \color{blue}{\left(0.918938533204673 + \left(-x\right)\right)} \]

      9. unsub-neg 100.0%

         \[\leadsto \left(\left(-y\right) \cdot 0.5 + x \cdot y\right) + \color{blue}{\left(0.918938533204673 - x\right)} \]

      10. associate-+r- 100.0%

          \[\leadsto \color{blue}{\left(\left(\left(-y\right) \cdot 0.5 + x \cdot y\right) + 0.918938533204673\right) - x} \]

      11. *-commutative 100.0%

          \[\leadsto \left(\left(\left(-y\right) \cdot 0.5 + \color{blue}{y \cdot x}\right) + 0.918938533204673\right) - x \]

      12. distribute-lft-neg-out 100.0%

          \[\leadsto \left(\left(\color{blue}{\left(-y \cdot 0.5\right)} + y \cdot x\right) + 0.918938533204673\right) - x \]

      13. distribute-rgt-neg-in 100.0%

          \[\leadsto \left(\left(\color{blue}{y \cdot \left(-0.5\right)} + y \cdot x\right) + 0.918938533204673\right) - x \]

      14. distribute-lft-out 100.0%

          \[\leadsto \left(\color{blue}{y \cdot \left(\left(-0.5\right) + x\right)} + 0.918938533204673\right) - x \]

      15. fma-def 100.0%

          \[\leadsto \color{blue}{\mathsf{fma}\left(y, \left(-0.5\right) + x, 0.918938533204673\right)} - x \]

      16. +-commutative 100.0%

          \[\leadsto \mathsf{fma}\left(y, \color{blue}{x + \left(-0.5\right)}, 0.918938533204673\right) - x \]

      17. sub-neg 100.0%

          \[\leadsto \mathsf{fma}\left(y, \color{blue}{x - 0.5}, 0.918938533204673\right) - x \]

   3. Simplified 100.0%

      \[\leadsto \color{blue}{\mathsf{fma}\left(y, x - 0.5, 0.918938533204673\right) - x} \]
   4. Step-by-step derivation

      1. fma-udef 100.0%

         \[\leadsto \color{blue}{\left(y \cdot \left(x - 0.5\right) + 0.918938533204673\right)} - x \]

      2. *-un-lft-identity 100.0%

         \[\leadsto \left(y \cdot \color{blue}{\left(1 \cdot \left(x - 0.5\right)\right)} + 0.918938533204673\right) - x \]

      3. *-un-lft-identity 100.0%

         \[\leadsto \left(y \cdot \color{blue}{\left(x - 0.5\right)} + 0.918938533204673\right) - x \]

      4. sub-neg 100.0%

         \[\leadsto \left(y \cdot \color{blue}{\left(x + \left(-0.5\right)\right)} + 0.918938533204673\right) - x \]

      5. metadata-eval 100.0%

         \[\leadsto \left(y \cdot \left(x + \color{blue}{-0.5}\right) + 0.918938533204673\right) - x \]

   5. Applied egg-rr 100.0%

      \[\leadsto \color{blue}{\left(y \cdot \left(x + -0.5\right) + 0.918938533204673\right)} - x \]

   6. Taylor expanded in x around inf 98.6%

      \[\leadsto \color{blue}{\left(y - 1\right) \cdot x} \]
   7. Step-by-step derivation

      1. *-commutative 98.6%

         \[\leadsto \color{blue}{x \cdot \left(y - 1\right)} \]

      2. sub-neg 98.6%

         \[\leadsto x \cdot \color{blue}{\left(y + \left(-1\right)\right)} \]

      3. metadata-eval 98.6%

         \[\leadsto x \cdot \left(y + \color{blue}{-1}\right) \]

      4. distribute-rgt-in 98.6%

         \[\leadsto \color{blue}{y \cdot x + -1 \cdot x} \]

      5. neg-mul-1 98.6%

         \[\leadsto y \cdot x + \color{blue}{\left(-x\right)} \]

      6. sub-neg 98.6%

         \[\leadsto \color{blue}{y \cdot x - x} \]

   8. Simplified 98.6%

      \[\leadsto \color{blue}{y \cdot x - x} \]

   if -0.76000000000000001 < x < 0.75

   1. Initial program 100.0%

      \[\left(x \cdot \left(y - 1\right) - y \cdot 0.5\right) + 0.918938533204673 \]
   2. Step-by-step derivation

      1. associate-+l- 100.0%

         \[\leadsto \color{blue}{x \cdot \left(y - 1\right) - \left(y \cdot 0.5 - 0.918938533204673\right)} \]

      2. sub-neg 100.0%

         \[\leadsto x \cdot \color{blue}{\left(y + \left(-1\right)\right)} - \left(y \cdot 0.5 - 0.918938533204673\right) \]

      3. metadata-eval 100.0%

         \[\leadsto x \cdot \left(y + \color{blue}{-1}\right) - \left(y \cdot 0.5 - 0.918938533204673\right) \]

   3. Simplified 100.0%

      \[\leadsto \color{blue}{x \cdot \left(y + -1\right) - \left(y \cdot 0.5 - 0.918938533204673\right)} \]

   4. Taylor expanded in x around 0 96.9%

      \[\leadsto \color{blue}{0.918938533204673 - 0.5 \cdot y} \]
3. Recombined 2 regimes into one program.

4. Final simplification 97.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -0.76 \lor \neg \left(x \leq 0.75\right):\\ \;\;\;\;x \cdot y - x\\ \mathbf{else}:\\ \;\;\;\;0.918938533204673 - y \cdot 0.5\\ \end{array} \]

Alternative 9: 100.0% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \left(0.918938533204673 + y \cdot \left(x + -0.5\right)\right) - x \end{array} \]

FPCore

(FPCore (x y) :precision binary64 (- (+ 0.918938533204673 (* y (+ x -0.5))) x))

C

double code(double x, double y) {
	return (0.918938533204673 + (y * (x + -0.5))) - x;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    code = (0.918938533204673d0 + (y * (x + (-0.5d0)))) - x
end function

Java

public static double code(double x, double y) {
	return (0.918938533204673 + (y * (x + -0.5))) - x;
}

Python

def code(x, y):
	return (0.918938533204673 + (y * (x + -0.5))) - x

Julia

function code(x, y)
	return Float64(Float64(0.918938533204673 + Float64(y * Float64(x + -0.5))) - x)
end

MATLAB

function tmp = code(x, y)
	tmp = (0.918938533204673 + (y * (x + -0.5))) - x;
end

Wolfram

code[x_, y_] := N[(N[(0.918938533204673 + N[(y * N[(x + -0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - x), $MachinePrecision]

Derivation

1. Initial program 100.0%

   \[\left(x \cdot \left(y - 1\right) - y \cdot 0.5\right) + 0.918938533204673 \]
2. Step-by-step derivation

   1. cancel-sign-sub-inv 100.0%

      \[\leadsto \color{blue}{\left(x \cdot \left(y - 1\right) + \left(-y\right) \cdot 0.5\right)} + 0.918938533204673 \]

   2. +-commutative 100.0%

      \[\leadsto \color{blue}{\left(\left(-y\right) \cdot 0.5 + x \cdot \left(y - 1\right)\right)} + 0.918938533204673 \]

   3. distribute-lft-out-- 100.0%

      \[\leadsto \left(\left(-y\right) \cdot 0.5 + \color{blue}{\left(x \cdot y - x \cdot 1\right)}\right) + 0.918938533204673 \]

   4. cancel-sign-sub-inv 100.0%

      \[\leadsto \left(\left(-y\right) \cdot 0.5 + \color{blue}{\left(x \cdot y + \left(-x\right) \cdot 1\right)}\right) + 0.918938533204673 \]

   5. *-rgt-identity 100.0%

      \[\leadsto \left(\left(-y\right) \cdot 0.5 + \left(x \cdot y + \color{blue}{\left(-x\right)}\right)\right) + 0.918938533204673 \]

   6. associate-+r+ 100.0%

      \[\leadsto \color{blue}{\left(\left(\left(-y\right) \cdot 0.5 + x \cdot y\right) + \left(-x\right)\right)} + 0.918938533204673 \]

   7. associate-+l+ 100.0%

      \[\leadsto \color{blue}{\left(\left(-y\right) \cdot 0.5 + x \cdot y\right) + \left(\left(-x\right) + 0.918938533204673\right)} \]

   8. +-commutative 100.0%

      \[\leadsto \left(\left(-y\right) \cdot 0.5 + x \cdot y\right) + \color{blue}{\left(0.918938533204673 + \left(-x\right)\right)} \]

   9. unsub-neg 100.0%

      \[\leadsto \left(\left(-y\right) \cdot 0.5 + x \cdot y\right) + \color{blue}{\left(0.918938533204673 - x\right)} \]

   10. associate-+r- 100.0%

       \[\leadsto \color{blue}{\left(\left(\left(-y\right) \cdot 0.5 + x \cdot y\right) + 0.918938533204673\right) - x} \]

   11. *-commutative 100.0%

       \[\leadsto \left(\left(\left(-y\right) \cdot 0.5 + \color{blue}{y \cdot x}\right) + 0.918938533204673\right) - x \]

   12. distribute-lft-neg-out 100.0%

       \[\leadsto \left(\left(\color{blue}{\left(-y \cdot 0.5\right)} + y \cdot x\right) + 0.918938533204673\right) - x \]

   13. distribute-rgt-neg-in 100.0%

       \[\leadsto \left(\left(\color{blue}{y \cdot \left(-0.5\right)} + y \cdot x\right) + 0.918938533204673\right) - x \]

   14. distribute-lft-out 100.0%

       \[\leadsto \left(\color{blue}{y \cdot \left(\left(-0.5\right) + x\right)} + 0.918938533204673\right) - x \]

   15. fma-def 100.0%

       \[\leadsto \color{blue}{\mathsf{fma}\left(y, \left(-0.5\right) + x, 0.918938533204673\right)} - x \]

   16. +-commutative 100.0%

       \[\leadsto \mathsf{fma}\left(y, \color{blue}{x + \left(-0.5\right)}, 0.918938533204673\right) - x \]

   17. sub-neg 100.0%

       \[\leadsto \mathsf{fma}\left(y, \color{blue}{x - 0.5}, 0.918938533204673\right) - x \]

3. Simplified 100.0%

   \[\leadsto \color{blue}{\mathsf{fma}\left(y, x - 0.5, 0.918938533204673\right) - x} \]
4. Step-by-step derivation

   1. fma-udef 100.0%

      \[\leadsto \color{blue}{\left(y \cdot \left(x - 0.5\right) + 0.918938533204673\right)} - x \]

   2. *-un-lft-identity 100.0%

      \[\leadsto \left(y \cdot \color{blue}{\left(1 \cdot \left(x - 0.5\right)\right)} + 0.918938533204673\right) - x \]

   3. *-un-lft-identity 100.0%

      \[\leadsto \left(y \cdot \color{blue}{\left(x - 0.5\right)} + 0.918938533204673\right) - x \]

   4. sub-neg 100.0%

      \[\leadsto \left(y \cdot \color{blue}{\left(x + \left(-0.5\right)\right)} + 0.918938533204673\right) - x \]

   5. metadata-eval 100.0%

      \[\leadsto \left(y \cdot \left(x + \color{blue}{-0.5}\right) + 0.918938533204673\right) - x \]

5. Applied egg-rr 100.0%

   \[\leadsto \color{blue}{\left(y \cdot \left(x + -0.5\right) + 0.918938533204673\right)} - x \]

6. Final simplification 100.0%

   \[\leadsto \left(0.918938533204673 + y \cdot \left(x + -0.5\right)\right) - x \]

Alternative 10: 73.5% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -8500000:\\ \;\;\;\;x \cdot y\\ \mathbf{elif}\;y \leq 1.1:\\ \;\;\;\;0.918938533204673 - x\\ \mathbf{else}:\\ \;\;\;\;x \cdot y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y -8500000.0)
   (* x y)
   (if (<= y 1.1) (- 0.918938533204673 x) (* x y))))

C

double code(double x, double y) {
	double tmp;
	if (y <= -8500000.0) {
		tmp = x * y;
	} else if (y <= 1.1) {
		tmp = 0.918938533204673 - x;
	} else {
		tmp = x * y;
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (y <= (-8500000.0d0)) then
        tmp = x * y
    else if (y <= 1.1d0) then
        tmp = 0.918938533204673d0 - x
    else
        tmp = x * y
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (y <= -8500000.0) {
		tmp = x * y;
	} else if (y <= 1.1) {
		tmp = 0.918938533204673 - x;
	} else {
		tmp = x * y;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= -8500000.0:
		tmp = x * y
	elif y <= 1.1:
		tmp = 0.918938533204673 - x
	else:
		tmp = x * y
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= -8500000.0)
		tmp = Float64(x * y);
	elseif (y <= 1.1)
		tmp = Float64(0.918938533204673 - x);
	else
		tmp = Float64(x * y);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= -8500000.0)
		tmp = x * y;
	elseif (y <= 1.1)
		tmp = 0.918938533204673 - x;
	else
		tmp = x * y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, -8500000.0], N[(x * y), $MachinePrecision], If[LessEqual[y, 1.1], N[(0.918938533204673 - x), $MachinePrecision], N[(x * y), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if y < -8.5e6 or 1.1000000000000001 < y

   1. Initial program 100.0%

      \[\left(x \cdot \left(y - 1\right) - y \cdot 0.5\right) + 0.918938533204673 \]
   2. Step-by-step derivation

      1. cancel-sign-sub-inv 100.0%

         \[\leadsto \color{blue}{\left(x \cdot \left(y - 1\right) + \left(-y\right) \cdot 0.5\right)} + 0.918938533204673 \]

      2. +-commutative 100.0%

         \[\leadsto \color{blue}{\left(\left(-y\right) \cdot 0.5 + x \cdot \left(y - 1\right)\right)} + 0.918938533204673 \]

      3. distribute-lft-out-- 100.0%

         \[\leadsto \left(\left(-y\right) \cdot 0.5 + \color{blue}{\left(x \cdot y - x \cdot 1\right)}\right) + 0.918938533204673 \]

      4. cancel-sign-sub-inv 100.0%

         \[\leadsto \left(\left(-y\right) \cdot 0.5 + \color{blue}{\left(x \cdot y + \left(-x\right) \cdot 1\right)}\right) + 0.918938533204673 \]

      5. *-rgt-identity 100.0%

         \[\leadsto \left(\left(-y\right) \cdot 0.5 + \left(x \cdot y + \color{blue}{\left(-x\right)}\right)\right) + 0.918938533204673 \]

      6. associate-+r+ 100.0%

         \[\leadsto \color{blue}{\left(\left(\left(-y\right) \cdot 0.5 + x \cdot y\right) + \left(-x\right)\right)} + 0.918938533204673 \]

      7. associate-+l+ 100.0%

         \[\leadsto \color{blue}{\left(\left(-y\right) \cdot 0.5 + x \cdot y\right) + \left(\left(-x\right) + 0.918938533204673\right)} \]

      8. +-commutative 100.0%

         \[\leadsto \left(\left(-y\right) \cdot 0.5 + x \cdot y\right) + \color{blue}{\left(0.918938533204673 + \left(-x\right)\right)} \]

      9. unsub-neg 100.0%

         \[\leadsto \left(\left(-y\right) \cdot 0.5 + x \cdot y\right) + \color{blue}{\left(0.918938533204673 - x\right)} \]

      10. associate-+r- 100.0%

          \[\leadsto \color{blue}{\left(\left(\left(-y\right) \cdot 0.5 + x \cdot y\right) + 0.918938533204673\right) - x} \]

      11. *-commutative 100.0%

          \[\leadsto \left(\left(\left(-y\right) \cdot 0.5 + \color{blue}{y \cdot x}\right) + 0.918938533204673\right) - x \]

      12. distribute-lft-neg-out 100.0%

          \[\leadsto \left(\left(\color{blue}{\left(-y \cdot 0.5\right)} + y \cdot x\right) + 0.918938533204673\right) - x \]

      13. distribute-rgt-neg-in 100.0%

          \[\leadsto \left(\left(\color{blue}{y \cdot \left(-0.5\right)} + y \cdot x\right) + 0.918938533204673\right) - x \]

      14. distribute-lft-out 99.9%

          \[\leadsto \left(\color{blue}{y \cdot \left(\left(-0.5\right) + x\right)} + 0.918938533204673\right) - x \]

      15. fma-def 99.9%

          \[\leadsto \color{blue}{\mathsf{fma}\left(y, \left(-0.5\right) + x, 0.918938533204673\right)} - x \]

      16. +-commutative 99.9%

          \[\leadsto \mathsf{fma}\left(y, \color{blue}{x + \left(-0.5\right)}, 0.918938533204673\right) - x \]

      17. sub-neg 99.9%

          \[\leadsto \mathsf{fma}\left(y, \color{blue}{x - 0.5}, 0.918938533204673\right) - x \]

   3. Simplified 99.9%

      \[\leadsto \color{blue}{\mathsf{fma}\left(y, x - 0.5, 0.918938533204673\right) - x} \]
   4. Step-by-step derivation

      1. fma-udef 99.9%

         \[\leadsto \color{blue}{\left(y \cdot \left(x - 0.5\right) + 0.918938533204673\right)} - x \]

      2. *-un-lft-identity 99.9%

         \[\leadsto \left(y \cdot \color{blue}{\left(1 \cdot \left(x - 0.5\right)\right)} + 0.918938533204673\right) - x \]

      3. *-un-lft-identity 99.9%

         \[\leadsto \left(y \cdot \color{blue}{\left(x - 0.5\right)} + 0.918938533204673\right) - x \]

      4. sub-neg 99.9%

         \[\leadsto \left(y \cdot \color{blue}{\left(x + \left(-0.5\right)\right)} + 0.918938533204673\right) - x \]

      5. metadata-eval 99.9%

         \[\leadsto \left(y \cdot \left(x + \color{blue}{-0.5}\right) + 0.918938533204673\right) - x \]

   5. Applied egg-rr 99.9%

      \[\leadsto \color{blue}{\left(y \cdot \left(x + -0.5\right) + 0.918938533204673\right)} - x \]

   6. Taylor expanded in x around inf 53.8%

      \[\leadsto \color{blue}{\left(y - 1\right) \cdot x} \]

   7. Taylor expanded in y around inf 53.2%

      \[\leadsto \color{blue}{y \cdot x} \]

   if -8.5e6 < y < 1.1000000000000001

   1. Initial program 100.0%

      \[\left(x \cdot \left(y - 1\right) - y \cdot 0.5\right) + 0.918938533204673 \]
   2. Step-by-step derivation

      1. associate-+l- 100.0%

         \[\leadsto \color{blue}{x \cdot \left(y - 1\right) - \left(y \cdot 0.5 - 0.918938533204673\right)} \]

      2. sub-neg 100.0%

         \[\leadsto x \cdot \color{blue}{\left(y + \left(-1\right)\right)} - \left(y \cdot 0.5 - 0.918938533204673\right) \]

      3. metadata-eval 100.0%

         \[\leadsto x \cdot \left(y + \color{blue}{-1}\right) - \left(y \cdot 0.5 - 0.918938533204673\right) \]

   3. Simplified 100.0%

      \[\leadsto \color{blue}{x \cdot \left(y + -1\right) - \left(y \cdot 0.5 - 0.918938533204673\right)} \]

   4. Taylor expanded in y around 0 95.2%

      \[\leadsto \color{blue}{0.918938533204673 + -1 \cdot x} \]
   5. Step-by-step derivation

      1. neg-mul-1 95.2%

         \[\leadsto 0.918938533204673 + \color{blue}{\left(-x\right)} \]

      2. sub-neg 95.2%

         \[\leadsto \color{blue}{0.918938533204673 - x} \]

   6. Simplified 95.2%

      \[\leadsto \color{blue}{0.918938533204673 - x} \]
3. Recombined 2 regimes into one program.

4. Final simplification 74.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -8500000:\\ \;\;\;\;x \cdot y\\ \mathbf{elif}\;y \leq 1.1:\\ \;\;\;\;0.918938533204673 - x\\ \mathbf{else}:\\ \;\;\;\;x \cdot y\\ \end{array} \]

Alternative 11: 50.2% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -4 \cdot 10^{-5}:\\ \;\;\;\;-x\\ \mathbf{elif}\;x \leq 0.92:\\ \;\;\;\;0.918938533204673\\ \mathbf{else}:\\ \;\;\;\;-x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x -4e-5) (- x) (if (<= x 0.92) 0.918938533204673 (- x))))

C

double code(double x, double y) {
	double tmp;
	if (x <= -4e-5) {
		tmp = -x;
	} else if (x <= 0.92) {
		tmp = 0.918938533204673;
	} else {
		tmp = -x;
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (x <= (-4d-5)) then
        tmp = -x
    else if (x <= 0.92d0) then
        tmp = 0.918938533204673d0
    else
        tmp = -x
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (x <= -4e-5) {
		tmp = -x;
	} else if (x <= 0.92) {
		tmp = 0.918938533204673;
	} else {
		tmp = -x;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if x <= -4e-5:
		tmp = -x
	elif x <= 0.92:
		tmp = 0.918938533204673
	else:
		tmp = -x
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (x <= -4e-5)
		tmp = Float64(-x);
	elseif (x <= 0.92)
		tmp = 0.918938533204673;
	else
		tmp = Float64(-x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= -4e-5)
		tmp = -x;
	elseif (x <= 0.92)
		tmp = 0.918938533204673;
	else
		tmp = -x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[x, -4e-5], (-x), If[LessEqual[x, 0.92], 0.918938533204673, (-x)]]

Derivation

1. Split input into 2 regimes

2. if x < -4.00000000000000033e-5 or 0.92000000000000004 < x

   1. Initial program 100.0%

      \[\left(x \cdot \left(y - 1\right) - y \cdot 0.5\right) + 0.918938533204673 \]
   2. Step-by-step derivation

      1. cancel-sign-sub-inv 100.0%

         \[\leadsto \color{blue}{\left(x \cdot \left(y - 1\right) + \left(-y\right) \cdot 0.5\right)} + 0.918938533204673 \]

      2. +-commutative 100.0%

         \[\leadsto \color{blue}{\left(\left(-y\right) \cdot 0.5 + x \cdot \left(y - 1\right)\right)} + 0.918938533204673 \]

      3. distribute-lft-out-- 100.0%

         \[\leadsto \left(\left(-y\right) \cdot 0.5 + \color{blue}{\left(x \cdot y - x \cdot 1\right)}\right) + 0.918938533204673 \]

      4. cancel-sign-sub-inv 100.0%

         \[\leadsto \left(\left(-y\right) \cdot 0.5 + \color{blue}{\left(x \cdot y + \left(-x\right) \cdot 1\right)}\right) + 0.918938533204673 \]

      5. *-rgt-identity 100.0%

         \[\leadsto \left(\left(-y\right) \cdot 0.5 + \left(x \cdot y + \color{blue}{\left(-x\right)}\right)\right) + 0.918938533204673 \]

      6. associate-+r+ 100.0%

         \[\leadsto \color{blue}{\left(\left(\left(-y\right) \cdot 0.5 + x \cdot y\right) + \left(-x\right)\right)} + 0.918938533204673 \]

      7. associate-+l+ 100.0%

         \[\leadsto \color{blue}{\left(\left(-y\right) \cdot 0.5 + x \cdot y\right) + \left(\left(-x\right) + 0.918938533204673\right)} \]

      8. +-commutative 100.0%

         \[\leadsto \left(\left(-y\right) \cdot 0.5 + x \cdot y\right) + \color{blue}{\left(0.918938533204673 + \left(-x\right)\right)} \]

      9. unsub-neg 100.0%

         \[\leadsto \left(\left(-y\right) \cdot 0.5 + x \cdot y\right) + \color{blue}{\left(0.918938533204673 - x\right)} \]

      10. associate-+r- 100.0%

          \[\leadsto \color{blue}{\left(\left(\left(-y\right) \cdot 0.5 + x \cdot y\right) + 0.918938533204673\right) - x} \]

      11. *-commutative 100.0%

          \[\leadsto \left(\left(\left(-y\right) \cdot 0.5 + \color{blue}{y \cdot x}\right) + 0.918938533204673\right) - x \]

      12. distribute-lft-neg-out 100.0%

          \[\leadsto \left(\left(\color{blue}{\left(-y \cdot 0.5\right)} + y \cdot x\right) + 0.918938533204673\right) - x \]

      13. distribute-rgt-neg-in 100.0%

          \[\leadsto \left(\left(\color{blue}{y \cdot \left(-0.5\right)} + y \cdot x\right) + 0.918938533204673\right) - x \]

      14. distribute-lft-out 100.0%

          \[\leadsto \left(\color{blue}{y \cdot \left(\left(-0.5\right) + x\right)} + 0.918938533204673\right) - x \]

      15. fma-def 100.0%

          \[\leadsto \color{blue}{\mathsf{fma}\left(y, \left(-0.5\right) + x, 0.918938533204673\right)} - x \]

      16. +-commutative 100.0%

          \[\leadsto \mathsf{fma}\left(y, \color{blue}{x + \left(-0.5\right)}, 0.918938533204673\right) - x \]

      17. sub-neg 100.0%

          \[\leadsto \mathsf{fma}\left(y, \color{blue}{x - 0.5}, 0.918938533204673\right) - x \]

   3. Simplified 100.0%

      \[\leadsto \color{blue}{\mathsf{fma}\left(y, x - 0.5, 0.918938533204673\right) - x} \]
   4. Step-by-step derivation

      1. fma-udef 100.0%

         \[\leadsto \color{blue}{\left(y \cdot \left(x - 0.5\right) + 0.918938533204673\right)} - x \]

      2. *-un-lft-identity 100.0%

         \[\leadsto \left(y \cdot \color{blue}{\left(1 \cdot \left(x - 0.5\right)\right)} + 0.918938533204673\right) - x \]

      3. *-un-lft-identity 100.0%

         \[\leadsto \left(y \cdot \color{blue}{\left(x - 0.5\right)} + 0.918938533204673\right) - x \]

      4. sub-neg 100.0%

         \[\leadsto \left(y \cdot \color{blue}{\left(x + \left(-0.5\right)\right)} + 0.918938533204673\right) - x \]

      5. metadata-eval 100.0%

         \[\leadsto \left(y \cdot \left(x + \color{blue}{-0.5}\right) + 0.918938533204673\right) - x \]

   5. Applied egg-rr 100.0%

      \[\leadsto \color{blue}{\left(y \cdot \left(x + -0.5\right) + 0.918938533204673\right)} - x \]

   6. Taylor expanded in x around inf 98.0%

      \[\leadsto \color{blue}{\left(y - 1\right) \cdot x} \]

   7. Taylor expanded in y around 0 48.2%

      \[\leadsto \color{blue}{-1 \cdot x} \]
   8. Step-by-step derivation

      1. neg-mul-1 48.2%

         \[\leadsto \color{blue}{-x} \]

   9. Simplified 48.2%

      \[\leadsto \color{blue}{-x} \]

   if -4.00000000000000033e-5 < x < 0.92000000000000004

   1. Initial program 100.0%

      \[\left(x \cdot \left(y - 1\right) - y \cdot 0.5\right) + 0.918938533204673 \]
   2. Step-by-step derivation

      1. associate-+l- 100.0%

         \[\leadsto \color{blue}{x \cdot \left(y - 1\right) - \left(y \cdot 0.5 - 0.918938533204673\right)} \]

      2. sub-neg 100.0%

         \[\leadsto x \cdot \color{blue}{\left(y + \left(-1\right)\right)} - \left(y \cdot 0.5 - 0.918938533204673\right) \]

      3. metadata-eval 100.0%

         \[\leadsto x \cdot \left(y + \color{blue}{-1}\right) - \left(y \cdot 0.5 - 0.918938533204673\right) \]

   3. Simplified 100.0%

      \[\leadsto \color{blue}{x \cdot \left(y + -1\right) - \left(y \cdot 0.5 - 0.918938533204673\right)} \]

   4. Taylor expanded in y around inf 98.5%

      \[\leadsto \color{blue}{y \cdot x} - \left(y \cdot 0.5 - 0.918938533204673\right) \]

   5. Taylor expanded in y around 0 48.6%

      \[\leadsto \color{blue}{0.918938533204673} \]
3. Recombined 2 regimes into one program.

4. Final simplification 48.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -4 \cdot 10^{-5}:\\ \;\;\;\;-x\\ \mathbf{elif}\;x \leq 0.92:\\ \;\;\;\;0.918938533204673\\ \mathbf{else}:\\ \;\;\;\;-x\\ \end{array} \]

Alternative 12: 27.1% accurate, 11.0× speedup

Math

\[\begin{array}{l} \\ 0.918938533204673 \end{array} \]

FPCore

(FPCore (x y) :precision binary64 0.918938533204673)

C

double code(double x, double y) {
	return 0.918938533204673;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    code = 0.918938533204673d0
end function

Java

public static double code(double x, double y) {
	return 0.918938533204673;
}

Python

def code(x, y):
	return 0.918938533204673

Julia

function code(x, y)
	return 0.918938533204673
end

MATLAB

function tmp = code(x, y)
	tmp = 0.918938533204673;
end

Wolfram

code[x_, y_] := 0.918938533204673

Derivation

1. Initial program 100.0%

   \[\left(x \cdot \left(y - 1\right) - y \cdot 0.5\right) + 0.918938533204673 \]
2. Step-by-step derivation

   1. associate-+l- 100.0%

      \[\leadsto \color{blue}{x \cdot \left(y - 1\right) - \left(y \cdot 0.5 - 0.918938533204673\right)} \]

   2. sub-neg 100.0%

      \[\leadsto x \cdot \color{blue}{\left(y + \left(-1\right)\right)} - \left(y \cdot 0.5 - 0.918938533204673\right) \]

   3. metadata-eval 100.0%

      \[\leadsto x \cdot \left(y + \color{blue}{-1}\right) - \left(y \cdot 0.5 - 0.918938533204673\right) \]

3. Simplified 100.0%

   \[\leadsto \color{blue}{x \cdot \left(y + -1\right) - \left(y \cdot 0.5 - 0.918938533204673\right)} \]

4. Taylor expanded in y around inf 73.9%

   \[\leadsto \color{blue}{y \cdot x} - \left(y \cdot 0.5 - 0.918938533204673\right) \]

5. Taylor expanded in y around 0 24.4%

   \[\leadsto \color{blue}{0.918938533204673} \]

6. Final simplification 24.4%

   \[\leadsto 0.918938533204673 \]

Reproduce

herbie shell --seed 2023274 
(FPCore (x y)
  :name "Numeric.SpecFunctions:logGamma from math-functions-0.1.5.2, A"
  :precision binary64
  (+ (- (* x (- y 1.0)) (* y 0.5)) 0.918938533204673))